Does enlightenment in all teachings amount to a progressively deeper realisation that I am already enlightened?

I was going to phrase this idea we are already enlightened as the present always already being past, which seems similar to Dogen's claim that enlightenment must be dropped every moment.

It seems that claim has links to 'original enlightenment' because the former is - at least superficially - the antithesis of the latter (it inverts the tense relation of enlightenment extending from the past to the present).

So no, according to Dogen we realise it is not - in practice - the case: forget what has been achieved.

Even as it may be abstractly.

Evolution in nature perpetuates samsara because it encourages becoming and it encourages reproduction. From the sutta quote below, you can see that going with the flow stands for craving. Craving is the cause of suffering according to the second noble truth. Evolution drives the natural tendency towards craving - sensual cravings and cravings towards becoming.

Going against the flow of nature is renunciation. Trying to end craving is against what evolution in nature drives us towards.

The idea of natural spiritual evolution is not found in (Theravada) Buddhism. The effort to becoming liberated is against natural evolution. Furthermore, as you can see in the Buddhist world today, extremely few join the monastic order in pursuit of liberation.

Natural intellectual evolution only leads towards increasing sensual enjoyment while attempting to reduce suffering, through improvement in healthcare, education, infrastructure, politics and economy.

Natural intellectual evolution does not lead towards liberation from clinging to sensual enjoyment. Rather, it leads one towards asking how to increase enjoyment while avoiding suffering as much as possible.

Furthermore, some minimum intelligence is required for becoming liberated, so that may exclude some people. Please see this answer for details.

However, with minimum intelligence, right effort and strong determination, anybody, regardless of their social background, has the equal opportunity to become fully liberated. Please see this answer for details.

Additionally, I think thinking that you're already enlightened but just need to uncover it, may be a kind of upaya (skillful means) to convince you to let go of trying too hard, while Iti 109 below employs upaya (skillful means) to convince you not to be too lax or lazy.

So, there's no right or wrong answer, you just need to get the right amount of effort as taught in Sona Sutta (AN 6.55) - about the strings of a lute not being too taut or too loose but just right.

From Iti 109:

This was said by the Blessed One, said by the Arahant, so I have heard: "Suppose a man was being carried along by the flow of a river, lovely & alluring. And then another man with good eyesight, standing on the bank, on seeing him would say: 'My good man, even though you are being carried along by the flow of a river, lovely & alluring, further down from here is a pool with waves & whirlpools, with monsters & demons. On reaching that pool you will suffer death or death-like pain.' Then the first man, on hearing the words of the second man, would make an effort with his hands & feet to go against the flow.

"I have given you this simile to illustrate a meaning. The meaning is this: the flow of the river stands for craving. Lovely & alluring stands for the six internal sense-media. The pool further down stands for the five lower fetters. The waves stand for anger & distress. The whirlpools stand for the five strings of sensuality. The monsters & demons stand for the opposite sex. Against the flow stands for renunciation. Making an effort with hands & feet stands for the arousing of persistence. The man with good eyesight standing on the bank stands for the Tathagata, worthy & rightly self-awakened."

Theravada Buddhism

There are four stages leading to attainment or Nirvana:-

1. Sothapanna - Stream-enterer
2. Sakadagami - Once-returner
3. Anagami - Non-returner
4. Arahath

In each stage, different elements of the ten sanyojana (fetters) will be dealt with. The Pali canon's Sutta Pitaka identifies ten fetters as:

1. belief in a self (Pali: sakkāya-diṭṭhi)
2. doubt or uncertainty, especially about the Buddha's awakeness and nine supermundane consciousnesses (vicikicchā)
3. attachment to rites and rituals (sīlabbata-parāmāsa)
4. sensual desire (kāmacchanda)
5. ill will (vyāpāda)
6. lust for material existence, lust for material rebirth (rūparāga)
7. lust for immaterial existence, lust for rebirth in a formless realm (arūparāga)
8. conceit (māna)
9. restlessness (uddhacca)
10. ignorance (avijjā)

-AN10.13 - Saṁyojanasutta

Fetters, 1-5 are known as lower fetters where-as fetters 6 - 10 are regarded as higher fetters. Lower fetters gets removed progressively in Sovan, Sakadagami and Anagami states, while the higher fetters are removed in an Arhath.

All who attain Nirvana can attain it via either of 3 ways in terms of effort required to attain Nirvana:

1. Savaka bodhi or Sravaka
2. Pacceka bodhi or Pachcheka buddha
3. Samma Sambodhi or Samma Sambudhdha

Sravaka bodhi requires the least effort to Nirvana, and thought of as the easiest and quickest path. Sravaka bodhi can only exist when there is Dhamma taught by Samma Sambudhdha is present in the world, as Sravaka (those who listen and learn Dhamma) can only exist when there is Dhamma present.

A Pachcheka buddha discovers enlightenment for himself, with own effort. A Pachcheka Buddha lacks the power to serve others by teaching Dhamma which he himself has discovered. Pachcheka Buddhas arise only during those periods when the teaching of a Samma Sambuddha does not exist. Pachcheka buddha spends more time than Shravaka in realizing Nirvana as it is done with own effort.

A Samma Sambuddha discovers enlightenment for himself, but in addition, he is able to teach Dhamma to others and show the path to Nirvana. Out of the 3 Bodhi, Samma Sambudhdha spends the most time. There are 3 categories of Samma Sambudhdha, in terms of the time required to attain Nirvana:

1. Praknknadhika bodhisathwa
2. Sadhdhadhika bodhisathwa
3. Wiiryadhika bodhisathwa

In the above list each bodhisathwa spends progressively more time in preparation to becoming budhdha. Least time is spent by Praknknadhika bodhisathwa, where most time is spent by Wiiryadhika bodhisathwa.

All three (Savaka bodhi, Pachcheka buddha, Samma Sambudhdha) are equal in terms of enlightenment, i.e. their liberation from sansara, which means that these are not three different classes of Nirvana.

However, the Sravaka and the Pachcheka bodhi are regarded as inferior to the Samma Sambuddha in terms of other, special qualities and capabilities. The arrival of Samma Sambudhdha is regarded as the highest form of mangala (luck) to all beings, because Samma Sambudhdha opens the path to Nirvana for many Sravaka.

The decision is left to the individual whether to take the way of the Sravaka, Pachcheka buddha or Samma Sambuddha.

EDIT: After the edit from the OP with more clarifications.

In Theravada, only the Arhath is enlightened. Sovan, Sakadagami and Anagami states can be thought of as intermediate states leading to Nirvana. I think there is nothing wrong in thinking of them as levels of progressive understanding. However, there is no way for the practitioner to know which state they are in. So, there is no such thing as realization of 'I'm already enlightened' for Sovan, Sakadagami or Anagami disciple. It is believed only Lord Budhdha can evaluate someone's state of mind correctly and declare the state (Sovan, Sakadagami or Anagami) they are in. However, one knows when they are enlightened, until their journey ends in Nirvana, disciples must strive towards Nirvana.

If by 'intrinsic enlightenment' you mean seeing through the world we have built around a 'self', yes it is Nirvana according to the Theravada Buddhism.

Apologies to OP, I am not qualified to compare Theravada with hongaku theory. I hope this answer will help you perform the comparison yourself.

"The second way in which mindfulness protects you is to keep you focused on what’s important. It reminds you that the most important thing that you have right now is the state of your mind, and if any unskillful thought comes in, your first priority is to get it out. The Buddha illustrates this point with an analogy of a man whose head is on fire. His mindfulness is focused on putting the fire out right away. He doesn’t simply watch the flames and note their pretty colors; he remembers that he’s got to put them out, and he can’t let anything else distract him. That’s the second way in which mindfulness helps protect your mind: It keeps your priorities in mind and keeps you focused on your task."

~ Thanissaro Bhikkhu "Mindfulness" https://www.dhammatalks.org/books/GoodHeart/Section0009.html

One experience of "already" is:

• Someone invites you to hear or read a description of Dharma
• When you're doing that they point out how clear your consciousness is

That's presented as an "intrinsic" clarity, and as attainable already/now.

But perhaps that's also "extrinsic" or conditioned, i.e. it's an awareness-of-clear-consciousness that's available when and because you're following a Dharma-talk.

To answer your question I suggest there could be three counter-arguments to the way you phrased it (i.e. that "every single enlightenment we can have in Buddhism is of our intrinsic enlightenment") -- by "counter-argument" I don't mean you're wrong but that misunderstanding or taking a wrong-sided view of the phrase could give a wrong impression:

• Some enlightenments are conditioned on skilful virtue -- see this answer -- you may choose to call that "intrinsic" but if it's entwined with how you relate with others then possibly "intrinsic" isn't the clearest adjective
• A statement like "I am enlightened" might be putting some undue (unhelpful?) emphasis on "I am" -- see this answer
• Similarly the phrase "our enlightenment" might be too possessive or self-aggrandizing -- see this answer ("not something we do, not our own practice").

I'm not sure what "Hongaku theory" is. Perhaps the important lesson from that theory isn't that "I am enlightened" but is more-so that "you are enlightened", as so for example:

• I shouldn't preach to you as if my enlightenment is superior to yours -- see this footnote about the Commentary to AN 5.159
• I shouldn't pity you -- see this topic
• I shouldn't assume that your leading a normal or mundane life implies you're stupid or unenlightened -- see this answer

Regarding common outcome, the Buddha says this:

MN121:13.1: Whatever ascetics and brahmins enter and remain in the pure, ultimate, supreme emptiness—whether in the past, future, or present—all of them enter and remain in this same pure, ultimate, supreme emptiness.

Regarding progression. the Buddha teaches that the skills developed in the progression are themselves transient.

MN121:11.3: They understand:
MN121:11.4: ‘Even this signless immersion of the heart is produced by choices and intentions.’
MN121:11.5: They understand: ‘But whatever is produced by choices and intentions is impermanent and liable to cessation.’

Most importantly, there is no "I am enlightened" or "I am already enlightened". Instead, the Buddha teaches:

MN121:12.1: They understand:
MN121:12.2: ‘Here there is no stress due to the defilements of sensuality, desire to be reborn, or ignorance.
MN121:12.3: There is only this modicum of stress, namely that associated with the six sense fields dependent on this body and conditioned by life.’
MN121:12.4: They understand: ‘This field of perception is empty of the perception of the defilements of sensuality, desire to be reborn, and ignorance.
MN121:12.5: There is only this that is not emptiness, namely that associated with the six sense fields dependent on this body and conditioned by life.’
MN121:12.6: And so they regard it as empty of what is not there, but as to what remains they understand that it is present.
MN121:12.7: That’s how emptiness is born in them—genuine, undistorted, and pure.

MN121 is an Early Buddhist Text. As such it predates Zen by a very long time. Studying both, one may see an unbroken connection as well as multiple cultural adaptations. Indeed, one may see the connection between emptiness and the need for Rinzai's katsu.

A monk asked: "What is the essence of Buddhism?"
The master gave a Katsu
The monk bowed.
The master said: "This one can hold his own in debate."

Teachings of Rinzai

Well... Yes, kind of... The essence of Teaching is the same in all schools and the Ultimate Attainment is one, what varies between the schools is the presentation. However, to tell someone just starting out: "you are already enlightened" would be rather misleading, even if true in a certain sense.

The essence of teaching in all schools is removal of conflict, towards the Ultimate Peace. This first involves removal of obvious causes of conflict, such as unwholesome behavior, and then removal of more tricky causes of conflict such as grasping at concepts and reifying abstractions.

When all causes of conflict are completely removed, the Ultimate Peace dawns automatically. Kinda makes obvious sense, right?

Now, different schools describe this ultimate peace in different ways. Some say that the Peace is ever present, and the conflict only exists in our minds. This is the "sun behind clouds" allegory. According to this school, Peace can be realized momentarily through Sudden Enlightenment - an insight liberating from all mental projection of conflict.

Other schools argue that the mind is not separate from the circumstances, and circumstances are fruits of past karma, so Peace cannot be attained without cleansing one's karma through a long and arduous process of Gradual Enlightenment.

Yet another group of schools claim, somewhat counterintuitively, that Ultimate Peace is the fundamental reality of all situations, including those that superficially seem like conflict. This view is known as The Great Perfection and goes hand in hand with the view of intrinsic enlightenment exemplified by the hongaku tradition.

It seems like the three types of schools are in disagreement but in reality they all speak about the same "elephant", emphasizing its different aspects. Indeed, we cannot progress on the path without removing at least some of the coarser conflict that comes from bad circumstances that came from problematic behavior. There's a point of diminishing return there though, beyond which trying to attain sterile circumstances with no conflict whatsoever only serves as a major source of inner conflict. This is when the schools teaching that "peace is a state of mind" suddenly seem more to the point. And once you practice it long and sincerely, you start getting the context in which "everything is Peace even when it's conflict" actually makes sense. The whole thing is a kind of progression.

Anyway, as you make progress weeding out conflict and cultivating peace, you start realizing that the Peace is attained right here and not in some fantasy land, hence the motto "samsara is nirvana".

Similarly, you start realizing that Peace is attained with wholesome behavior and pure mind, not with some magical superpowers. You realize there's nothing you have been lacking for the proper practice of peace here and now. You just have to do it. Hence the motto, "you are already enlightened".

All schools of Buddhism know this truth, the situation we are in. It's just that some of them describe it from the perspective of a beginner barely familiar with Buddhism, others speak to an intermediate student's view, and the non-dualist schools operate within an enlightened perspective.