Why do enlightened beings want to save sentient beings? Does everyone want to be saved?

More than one Buddha has faced this question. Here there are many many beings, with much dust in their eyes, who choose to chase pleasure and run away from suffering. With so many having dust in their eyes, new Buddhas may initially be inclined to do nothing:

DN14:3.2.1: And then these verses, which were neither supernaturally inspired, nor learned before in the past, occurred to him: ‘I’ve struggled hard to realize this, enough with trying to explain it! This teaching is not easily understood by those mired in greed and hate. Those caught up in greed can’t see what’s subtle, going against the stream, deep, hard to see, and very fine, for they’re shrouded in a mass of darkness.’ So, as the Buddha Vipassī reflected like this, his mind inclined to remaining passiv e, not to teaching the Dhamma. Then a certain Great Brahmā, knowing what the Buddha Vipassī was thinking, thought, ‘Oh my goodness! The world will be lost, the world will perish! For the mind of the Realized One Vipassī, the perfected one, the fully awakened Buddha, inclines to remaining passive, not to teaching the Dhamma.’ Then, as easily as a strong person would extend or contract their arm, he vanished from the Brahmā realm and reappeared in front of the Buddha Vipassī.

Yet at divine invitation, such Buddhas will look more carefully to see that some people have little dust in their eyes.

DN14:3.3.1: He arranged his robe over one shoulder, knelt on his right knee, raised his joined palms toward the Buddha Vipassī, and said, ‘Sir, let the Blessed One teach the Dhamma! Let the Holy One teach the Dhamma! There are beings with little dust in their eyes. They’re in decline because they haven’t heard the teaching. There will be those who understand the teaching!’

Looking around, all these Buddhas, out of compassion, then decided to teach those who would listen. Those who would not listen now might listen later. Indeed Moggallana did not heed Buddha Kakusandha, and went to hell for an eon before he had ears to hear Buddha Gautama.

MN50:8.1: Once upon a time, Wicked One, I was a Māra named Dūsī, and I had a sister named Kāḷī.

From these suttas, one can understand that some can be helped, yet not all can be saved here and now even by a single Buddha.

Please do not torment yourself. Be at peace and may your practice be fruitful.

There were silent Buddhas (paccekabuddha) who did not help others, but only attained liberation for themselves. They had compassion but they did not have the capability to help the world at large.

On the other hand, Gautama Buddha was a sammasambuddha, who was not only enlightened but had both compassion and the capability to help the world at large.

The noble and the conventional sangha of course carries on the propagation and practice of the Buddha's teachings.

As you have rightly surmised, to go with the flow of nature and biology is to go in the direction of embracing craving and clinging, and enjoying sensual pleasures, and trying to become something. Nature and biology also brings the evolutionary drive to reproduce, and minimize suffering, while maximizing sensual enjoyment.

Suffering or dukkha has two meanings, which you can find in this answer. The superficial meaning is literal pain and suffering. And the deeper meaning is unsatistactoriness or discontent. Going with the flow of nature and biology, which is going with the flow of craving and clinging, will never result in stable happiness and security. It will always lead to the general feeling of unsatisfactoriness.

The enlightened beings are compassionate because they see others suffering in the manner described, and want to give them the help they need to become liberated from suffering, but they realize that not everyone wants to become free.

Most of humanity is too deeply entrenched in sensual pleasures and experience the vicious cycle of stimulating craving for more sensual pleasures. The appetite for sensual pleasures increase endlessly. That temporarily masks suffering, but in the long term, suffering is inevitable. So, most of humanity is not interested in liberation from suffering, even if they experience unsatisfactoriness most of the time.

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."

From Magandiya Sutta:

"Now suppose that there was a leper covered with sores & infections, devoured by worms, picking the scabs off the openings of his wounds with his nails, cauterizing his body over a pit of glowing embers. The more he cauterized his body over the pit of glowing embers, the more disgusting, foul-smelling, & putrid the openings of his wounds would become, and yet he would feel a modicum of enjoyment & satisfaction because of the itchiness of his wounds. In the same way, beings not free from passion for sensual pleasures — devoured by sensual craving, burning with sensual fever — indulge in sensual pleasures. The more they indulge in sensual pleasures, the more their sensual craving increases and the more they burn with sensual fever, and yet they feel a modicum of enjoyment & satisfaction dependent on the five strings of sensuality.

Most people are like heroin addicts always looking for that next fix. There is a far greater wellbeing that we can't know unless our habit of being distracted by sensual desires are put aside or destroyed permanently. You lack context as we all do of what it truly means to be happy and at peace.

What happens when you die? What happens when you get sick? What happens when your family dies? What happens when a burglar has broken into your house? What if someone ran over your arm with a steamroller? Can you handle those things without reacting with the habit of anger and greed that cloud the mind? The Buddha's Teaching teaches us how to transcend these sufferings.

This question is beyond excellent, thank you for asking it!

It boils down to potential and access. Every existence has the potential to reach enlightenment; the premise behind your question comes down to opportunity of choice. If there is no access for one to pursue enlightenment then the potential of the individual in question is for naught.

The action of creating access through opportunity comes down to desire. If someone has lived a difficult path and observe another who is not affected by the same suffering; then the opportunity to learn from the one not suffering is not beyond reproach.

What people consider as a hell is variable in definition, dependant on the individual you talk to; but hell can defined as a position where hope or compassion is not present. If someone believes they deserve the squalor of the hell they live in, they are allowed to continue to live in such conditions. However, one decision of one person should not affect the next.

The premise of enlightenment is basically the control of our own environment. Suffering through the turbulence of another's situation only creates turbulence in our own situation; which in turn repeats the cycle with the centre of the suffering just shifting from individual to individual (if we let it). The process of becoming enlightened is learning how not to be affected by the suffering of others, which creates a sense of harmony around the individual and if others are interested in getting on the enlightened one's 'level', the opportunity to do so is within accessibility.

Finally, say humanity managed to solve all the issues of suffering, where to go next? Human's are not the only individuals on this planet, let alone the universe. If an individual cries for help; ignoring the Dharma of the Lady of Compassion is ignoring the Dharma and so, the cycle of suffering will relapse and continue until the Dharma of all individuals has be acknowledged.

As Plato expressed, "Ignorance is the root of all evil."

Cultivate in harmony friend

First, it is not a state of nothingness, but rather than the squalor of individual aloneness, in which each of us must struggle and fight for our own survival in a world that doesn’t care whether we live or die, there is the realization of non-separateness—that every being is not an ‘other’ because there are no independent existences anywhere, not even our own, no more than the fingers of your hand are separate beings, each with its own destiny and struggle. This is called Emptiness because there are no things, or beings, that are independent of the primordial ground—Buddha nature. Thus it is “not thingness” rather than “nothingness”.

Second, enlightened beings are not compassionate in the manner that unenlightened beings are compassionate, who, if they have any compassionate feelings towards other beings at all, do so from a deluded sense of being an individual self acting to help another. It is a feeling hopefully connected to an action done by them and for another. An enlightened being who has realized that there is no separate me or you has no choice but to respond as if—as we would normally think of it in our ignorance—the suffering is their own, and as “instinctively” as an unenlightened being acts through the delusion of ignorance to their own profit, gain, or happiness. This is Great Compassion (Mahākarunā). There is no aspiration or striving to be this way, because there is no other way to be, once full realization is accomplished.

Third “suffering” is more than just bodily illness or injury. Today in a world blighted by a pandemic, it is not just those who are suffering from the disease, but those who fear the loss of their home or job, who do not have the freedom to move about, who can not realize a dream long held, or see their family, or friends, etc. We suffer from frustration, from fear, from doubt, from anger, from anxiety, and from loss much more often than we actually suffer physically. And if you are suffering in this way, you feel an overriding desire to free yourself from this suffering. An enlightened being doesn’t look compassionately upon this suffering because there is no ‘other’ that is suffering it ‘themselves”—there is just suffering and they must act. No choice in the matter.

And every hero stops worrying about what they have to do and just does it, and this is true of an enlightened being even more so. This is tranquillity—accepting what is, rather than being attached to what was, or yearning for what could otherwise be instead.

I see this as the only path worth walking. The rest are based in ignorance of our true nature, and of the beauty and harmony of a world empty of personal aggrandizement at the expense of everyone else. This path is rooted in an understanding of beauty, love, and inseparable destiny. Who wouldn’t wish that for their world?

If you knew of a really good restaurant, would you keep that knowledge to yourself, or would you take your friends there? If you saw a fantastic movie, would you go home and muse about it in private, or would you try to get your acquaintances to go see it? There's nothing mystical or magical here. People like to share good things with others; sharing knowledge of what's good creates value without cost. Why wouldn't one want to do it?

Now, people who lack understanding can get contracted. They worry that if they share knowledge of a good thing, it will create demand, then shortages. The thought of sharing the knowledge of something good invokes a fear of loss; the fear of losing something good invokes an urge to grasp; soon such people are trying to stockpile what's good, to decide who is worthy to acmes it and who is not, to police the boundaries of what they have determined is 'theirs' with all necessary force. The 'good' becomes like the ancient myth of the dragon's hoard: an accumulation of value for no purpose save to stave off that fundamental and abstract fear of loss and magnify the ego.

Are such people happy? Well, I suppose yes, but only in the twisted sense the they can share the knowledge of their own superiority with others: "I have, and you don't"; "we can do this, and they can't"; "our people are worthy, theirs are not". Their suffering isn't 'misery' in the classic sense, but more that they have no real understanding of their own selves, Like Milton's famous passage:

And they, so perfect is their misery,
Not once perceive their foul disfigurement,
But boast themselves more comely then before
And all their friends, and native home forget
To roule with pleasure in a sensual sty. — Comus, lines 73-7

Without that comparative — without the idea that 'the good' is in short supply, and that one has beaten others to secure it for oneself — this kind of wealth has no value, and makes no one happy at all.

Enlightenment is freeing — it frees us from the enchantment of that 'hoard' mentality, and the web of cravings that surround it — but it makes one conscious of the fetters that others are subject to. It's bittersweet. But still, it's a good that anyone who finds it would like to share, if they can figure out how.