Why is 'dukkha' included in one of the three marks of existence?

It might be worth mentioning that in English they're called "the three marks of existence" -- but in Pali the word tilakkhaṇa literally means the "three marks" or "three characteristics" (not "of existence"), or signata -- and the "things" which have (or which are ascribed) the dukkha characteristic are specifically saṅkhāras.

The three marks are:

• sabbe saṅkhārā aniccā — "all saṅkhāras (conditioned things) are impermanent"
• sabbe saṅkhārā dukkhā — "all saṅkhāras are unsatisfactory"
• sabbe dhammā anattā — "all dharmas (conditioned or unconditioned things) are not self"

"saṅkhāra" (and dhamma) is a complicated word, see Can anyone explain Sanskara / Sankara indepth?

See also Did the Buddha really say that "life is suffering"? -- and, this answer.

I guess this agrees with what you were saying in the question, i.e. that suffering is not an essential characteristic of "existence" (i.e. life) -- what it is, is a characteristic of conditioned/compound things.

See also this -- after knowledge (ñāṇa) of "unsatisfactoriness" there's knowledge of "disenchantment" (nibbida) ...

7. Adinava ñana - Knowledge of mental and physical states as dukkha. "So he sees, at that time, only suffering, only unsatisfactoriness, only misery." [7]
8. Nibbida ñana - Knowledge of disenchantment/disgust with conditioned states.

... which I think is an important aspect of the soteriology of the Pali canon, and of post-canonical Theravada.

Perhaps there's similar/related doctrine in Zen, e.g. "no attachment to dust" and so on (or perhaps not exactly or not only "dust", referring to physical object, but especially the eight worldly winds, which I think are a doctrine known to all schools).

And beginning to read about nibbida, it's about conditioned mental states at least as much or more than it is about conditioned things (which I think parallels ruben2020's and MAGA2020's answers).

I’d start with what the Buddha said was the goal of life: it is, simply, “dwelling, here and now, beyond appetites, consummate, unfevered, in bliss, in (wholesomeness).” (Further Dialogues of the Buddha, Vol. I, p. 247, Tr. Lord Chalmers, Oxford University, London 1926) So that: “... to whatever place you go, you shall go in comfort; wherever you stand, you shall stand in comfort; wherever you sit, you shall sit in comfort; and wherever you make your bed, you shall lie down in comfort.” (The Book of the Gradual Sayings, Vol IV, p. 200, Tr. E.M. Hare, Luzac and Co., Ltd., London, 1955.

Note that the goal of life is neither more favorable rebirth, nor extinction of self or of desire, as some maintain. For life without desire is not living, living only for some future benefit is to ignore the compassionate love and beauty that is everywhere around us, and extinction of self is not possible.

The Buddha’s great insight, which he rarely stated as the “four noble truths,” but rather more succinctly as the insight that: Desire for what will not be attained ends in frustration (which is suffering); therefore, to avoid frustration, avoid desiring what will not be attained. This then gets expanded into the four noble truths, which is a more explicit version of the Buddha’s insight for those that cannot work out the significance of the simpler version.

Certainly, desiring food when hungry, water when thirsty, and shelter from inclement weather, companionship when lonely, etc., do not cause suffering when they are fulfilled. But what if you desire a five-course meal at a Michelin-starred restaurant, but can only afford a salad at home? If you cannot accept what you have, you will be frustrated and therefore suffer. And, of course, today we live in sociopathic cultures that barrage us all day long with things beyond our reach, to desire, so certainly in this sense alone, dukkha is a quality of our lives.

But even the desire to not desire more than you will obtain, or have, in your life, is a subtle desire that may not be obtained—-at least until you have sufficiently trained your mind to no longer be swayed by all the tantalizing carrots that are held just beyond your grasp. So, before that stage, you must be compassionate towards yourself and others, seeing that we are all initially frail and subject to dropping the ball at times, and desiring more than we can deliver—-in this case the subtle desire to be free from frustrated desires.

So the intermediate goal is to give up trying to solve the problem of frustration, by becoming willing to accept what you have—-accepting what is and not wishing it was different.

Remembering that we cannot know what we will get, leaves us room to motivate ourselves by desiring to achieve something, such as having a trained mind, or getting a certain job; but always accepting what comes with a glad heart.

Like trying to extinguish the self, which is just illusory, and thus, nothing that can be extinguished, but rather the illusory self just dissolves when we see through that illusion—-frustration that leads to suffering is not something we can extinguish by trying to suppress desire. Instead, frustration leading to suffering will just dissolve once we can gladly accept what we have in each and every moment.

And because going against the natural impulses is difficult to impossible to accomplish without proper intent and training, frustration and thus suffering are qualities of our everyday lives.

All feelings, perceptions & existence are dukkha because they are an expression of the conditioned element.

There are two elements to be known; conditioned and the unconditioned.

Not knowing the unconditioned which comes to be known through it's realization as the 'nibbananirodhadhatu' [extinguishment-cessation-principle] of the conditioned [dukkha].

The target referent of "cessation of suffering" is the principle by which suffering ends.

It is just that and it is a truth to be attained and seen as it actually is.

It does not belong to anyone nor is it an inherent property of suffering. Suffering is one thing, it's end is another.

Extinguishment of a fire is not an eternal fire, the principle of it's extinguishment is not an inherent property of a flame or it's fuel, analogically the extinguishment of perceived existence isn't an eternal perception of bliss because it's an release from perception itself. It is the end of dukkha, a release & a freedom from dukkha.

I will give an analogy; "If very cruel people conceived a child and raised him as a cruel joke to only know abuse, they would treat him wrong, train him wrong all whilst teaching we are treating you good, you are lucky and we are training you correctly.

Then the child not knowing better might think;'I am so fortunate, my masters are kind to me, they treat me good and are showing me kindness'.

If he were to escape however and learn the truth, he would be horrified by his former existence."

Likewise when one sees the truth of cessation it becomes known as the primary bliss.

Therefore as there are two elements; the conditioned and the unconditioned. There are are two elements to be known; dukkha and happiness.

Here sutta excerpts;

There are, Ānanda, these two elements: the conditioned element and the unconditioned element.

Bhikkhus, there are these three characteristics that define the conditioned. What three? An arising is seen, a vanishing is seen, and its alteration while it persists is seen. These are the three characteristics that define the conditioned.

The born, come-to-be, produced, The made, the conditioned, the transient, Conjoined with decay and death, A nest of disease, perishable, Sprung from nutriment and craving's cord — That is not fit to take delight in.

There are, Ānanda, these two elements: the conditioned element and the unconditioned element.

“Bhikkhus, there are these three characteristics that define the unconditioned. What three? No arising is seen, no vanishing is seen, and no alteration while it persists is seen. These are the three characteristics that define the unconditioned.”

"There is, bhikkhus, a not-born, a not-brought-to-being, a not-made, a not-conditioned. If, bhikkhus, there were no not-born, not-brought-to-being, not-made, not-conditioned, no escape would be discerned from what is born, brought-to-being, made, conditioned. But since there is a not-born, a not-brought-to-being, a not-made, a not-conditioned, therefore an escape is discerned from what is born, brought-to-being, made, conditioned." The escape from that [from the conditioned], the peaceful, beyond reasoning [beyond objectification], Constant, The not-born, the unproduced, The sorrowless state that is void of stain, The cessation of states associated with suffering, The stilling of the conditioned — bliss.

There is, bhikkhus, that base where there is no earth, no water, no fire, no air; no base consisting of the infinity of space, no base consisting of the infinity of consciousness, no base consisting of nothingness, no base consisting of neither-perception-nor-non-perception; neither this world nor another world nor both; neither sun nor moon. Here, bhikkhus, I say there is no coming, no going, no staying, no deceasing, no uprising. Not fixed, not movable, it has no support. Just this is the end of suffering.

'Three feelings have been taught by the Blessed One: pleasant, painful and neutral feelings. But the Blessed One has also said that whatever is felt is within suffering.' Now, with reference to what was it stated by the Blessed One that whatever is felt is within suffering?" "Well spoken, monk, well spoken! While three feelings have been taught by me, the pleasant, the painful and the neutral, yet I have also said that whatever is felt is within suffering. This, however, was stated by me with reference to the impermanence of (all) conditioned phenomena.[1] I have said it because conditioned phenomena are liable to destruction, to evanescence, to fading away, to cessation and to change. It is with reference to this that I have stated: 'Whatever is felt is within suffering.'

202. There is no fire like lust, no crime like hate. There is no ill like the body, no bliss higher than Peace (Nibbāna)
203. Hunger is the greatest disease. Aggregates are the greatest ill. Knowing this as it really is, (the wise realize) Nibbāna, bliss supreme.
204. Health is the highest gain. Contentment is the greatest wealth. The trusty are the best kinsmen. Nibbāna is the highest bliss. (Dhammapada verses)

Conditioned existence arises as one thing and ceases as another. Past is one end, future is another end and present in the middle. The cessation of the conditioned is an escape from this everchanging existence.

The pleasant feelings we encounter they are merely bait, there is a snake in the grass and they are compared to borrowed goods.

They are Mara's bait, even a little craving will incline the mind to becoming.

Want a pleasant experience? Well here good sir, take this birth of a body, subject to disease, old age and death. Go on to experience that love, or sensual pleasure and be parted with it, what a cruelty(!). What on earth have we done to deserve being thus crushed again and again?!

Spending most of our lives looking for a footing just to have it pulled out from underneath and falling.

Seeing these flaws, some might say 'Well that's just how it is, deal with it'. So speak the victims of abuse not knowing better.

It's a horrible deal, a trap. Wanting what is pleasant we go along as if we have no choice, tied to old age, sickness and death. As a dumb starving mouse would go for the cheese in a mousetrap perceiving pleasantness in suffering.

We are truly all being abused and we should not compromise our happiness. We should battle Mara with sword of insight and win the unconditional joy because we deserve it and because we can.

It is the usual dialogues between the buddha and the monks

“Monks, is bodily form permanent or impermanent?”

The monks said to the Buddha: “It is impermanent, Blessed One.”

The Buddha said: “Monks, what is impermanent, is it dukkha?”

The monks said to the Buddha: “It is dukkha, Blessed One.”

The Buddha said: “Monks, what is impermanent, dukkha, of a nature to change, would a learned noble disciple herein regard it as the self, as distinct from the self in the sense of being owned by it, as existing within the self, or the self as existing within it?“

The monks said to the Buddha: “No, Blessed One.”

The Buddha said: “Feeling … perception … formations … consciousness are also like this. Therefore, monks, whatever bodily form, whether past, future or present, internal or external, gross or subtle, sublime or repugnant, far or near, all of it is entirely not self, not distinct from the self in the sense of being owned by it, does not exist within the self, nor does a self exist within it. Examine it as it really is. Feeling … perception … formations … consciousness are also like this.

“A learned noble disciple attains liberation from bodily form, attains liberation from feeling … perception … formations … consciousness. I say, he is liberated from birth, old age, disease, death, worry, sorrow, vexation, and pain, from this entire great mass of dukkha.’”

https://suttacentral.net/sa87/en/analayo

The general statement is this

Profound indeed is this, namely conditioned genesis; even more profound, more difficult to see (perceive) is this, namely the extinction of all attachment, the destruction of craving, the fading away of desire, cessation: nirvāṇa.

“These two dharmas are namely the compounded and the uncompounded. The compounded is arising, persisting, changing, passing away. The uncompounded is not arising, not persisting, not changing, not passing away.

“Monks, this is to say: All activities, compounded things are suffering, and their cessation is nirvāṇa. When the causes are there, suffering arises; when the causes cease, the suffering ceases.

“All routes are cut off, the continuum ceases. The cessation of the continuum is called the ending of suffering.

“O monks! What is it that ceases? It is any remaining suffering. If this ceases, that is coolness, tranquillity, namely the extinction of all attachment, the destruction of craving, the fading away of desire, cessation: nirvāṇa.” https://suttacentral.net/sa293/en/choong

First, I'd be a bit careful about ascribing a causal relationship between dukkha and tanhā ('discontentment' and 'craving'). If I remember correctly there is a long-running debate among scholars about which causes the other, and in any case it strikes me the two represent different dimensions of analysis: a passive state and an active attitude. "I am discontent" is a different mode of thought and behavior from "I feel a craving".

To the question, dukkha is a result of the relationship between the thinking mind and the world. Literally, we have dukkha because we can imagine things otherwise. If I have a hamburger, I am only discontent if I start thinking about how I could have had a nice juicy steak; if I have no food at all I might be hungry, but I can't be discontent unless I imagine what it feels like to be full. As humans we have powerful capacities to imagine otherwise — it's natural for us to do so — and so we have dukkha.

Tanhā is more of an obstinate desire to remove that discontentment (dukkha) by force of will, a desire that (in part through its sheer pigheadedness) creates further discontentment in ourselves and others. It's a cycle: we feel discontent, so we try to force the universe into a pattern that pleases us, and end up producing more discontentment.

So now we get to the meat of the matter (the hamburger, as it were).

• sabbe saṅkhārā aniccā: everything is impermanent. Things we like come and go; things we don't like come and go; things we don't care about either way come and go.
• sabbe saṅkhārā dukkhā: therefore, everything makes us discontent. We want what we like to come and stay; we want what we dislike to go and stay away; we want what we don't care about to stop bothering us or to become better. That doesn't happen, not in any permanent sense, and we find that frustrating.
• sabbe dhammā anattā: all right teaching (dhammā) leads us away from this imagining/imagined self. When we think about eating a nice juicy steak and feeling deliciously full, we are imagining ourselves (as though we were another person) eating that juicy steak and feeling deliciously full. We get jealous of that other self; we want to be him/her, and we worry that we never will. But if we remove that imagined other self, whom do we have to be jealous of? What do we have to be discontent about?

So the real issue boils down to insight and perspective. Is our discontent due to something we can address and resolve? Or is it a hamster wheel where we are always trying to get to some happier self, never realizing that the wheel is turning away under our feet? Is our joy a natural, organic upwelling in the moment? Or is it a cruel joy we feel because we've beaten out some imagined 'loser' self that would have done worse than we did? The second of each of these will always produce dukkha, because we are seeing ourselves in relation to that imagined self.

The teachings of the Buddha are not metaphysics. They do not describe the ontology or nature of things.

The teachings of the Buddha serve one purpose only - freedom from suffering. It's purpose is soteriological, not ontological.

In "Anicca Vata Sankhara", Ven. Bodhi comments:

The most important fact to understand about sankharas, as conditioned formations, is that they are all impermanent: "Impermanent, alas, are formations." They are impermanent not only in the sense that in their gross manifestations they will eventually come to an end, but even more pointedly because at the subtle, subliminal level they are constantly undergoing rise and fall, forever coming into being and then, in a split second, breaking up and perishing: "Their very nature is to arise and vanish." For this reason the Buddha declares that all sankharas are suffering (sabbe sankhara dukkha) — suffering, however, not because they are all actually painful and stressful, but because they are stamped with the mark of transience. "Having arisen they then cease," and because they all cease they cannot provide stable happiness and security.

First we have the first noble truth that tells us that there is suffering or dukkha. This is explained in this answer.

Next, we see - sabbe sankhara anicca.

This mark of existence states that all conditioned and compounded things (both mental and physical) are impermanent.

So what? What is the connection to you?

Next, we see - sabbe sankhara dukkha.

This mark of existence states that because all conditioned and compounded things (both mental and physical) are impermanent, therefore they are a source of suffering or unsatisfactoriness, because "they cannot provide stable happiness and security." They are connected in this way.

These two marks of existence are there to show the connection from all conditioned and compounded things to the first noble truth of dukkha. In particular, they show the connection of the impermanence of the five aggregates (which are conditioned and compounded) to dukkha.

It's not there to explain the nature of reality. It's there as part of the narrative to explain the four noble truths.

The three marks of existence are mainly about YOUR existence. The marks of existence of a chair or table is unimportant.