Percy is the Child of Prophecy, more of Prophecy of Big Three Child than Prophecy of the Seven. Part because in the former, he shoulders it mostly alone, while in the later, he shares the burden with six other demigods and they know it. In the latter, while his choice and actions do matter, it was not end of all. The others' action and choices in turn also helped in shouldering the prophecy to achieve its best outcome (for Demigods and the Gods) while in the former, Percy bear the brunt of it, and he is the only one that can bring the best outcome to the Olympus.
Let's see… Percy's characteristic.
He is sarcastic, sassy, and picking fight with immortals without even meaning to. But, he also loyal, street-smart, and having his moment of empathy even to his enemies, in that he understands other's point even if he didn't like it. He cannot make long-term plan (that actually works) but he can adapt fast and when he makes a commitment, he tries his best.
This is why him being bearer of the First Great Prophecy is a Good Thing for the Olympians. Because to earn their Good Ending, the Child of Prophecy is not supposed to be just viewing the gods as All Powerful Being That Can Do No Wrong; but someone who understands their points and respect it, even if begrudgingly. Someone who is not afraid to point out Olympians' Flaw to their face without being too revolting about it.
Now, let's consider this alternative: What if Percy Jackson does not exist to the Story?
Perhaps he never born. Perhaps he was never found by the satyr. Perhaps a monster get to him first. Perhaps one of the gods figure out his identity and kill him. Either way, Percy Jackson does not exist, or if he does, he does not contribute to The Story. Does Not do the Feats.
How would it affects the Olympians?
Easy. Bad Ends.
Bad End 1: The Lightning Thief. Luke somehow successfully sends the Master Bolt to Kronos. Even if he didn't, the Bolt and the Helmet are still missing; and Ares is still under Kronos' control. This give the Titan a "Sleeper Agent" among the Olympian Council and can subtly influence their decision to benefit them, while the Gods wage Three-Way War. Even if a Quest is issued, what can guarantee them to find the Items on time and return it? Even if we taking Echidna out of the equation (because Zeus send her to Percy specifically), there are still a lot of danger. Perhaps the Fury get into them on the Bus. Perhaps Medusa turned them into statues. Perhaps they failed to get Ares' Shield. Perhaps they are STILL stuck in the Lotus Hotel. Perhaps they cannot leave the Underworld. Either way, the Quest failed. The war rages on.
Bad End 2: assuming they manage to avoid disaster that was The Lightning Thief Incident by sheer miracle, there are Sea of Monster and Thalia Tree getting poisoned. Who would come to the camp knowing the exact location to the Polyphemus Island? Who will force Tantalus to issue the Quest? Who will Hermes tasked and can they do it? Assuming the Questers manage to get INTO the Sea of Monster and get to the Polyphemus Island to get the Golden Fleece, can they leave? Alive? When the Ocean is against them? With Princess Andromeda is waiting on them at Miami Shore? Kronos would get the Fleece and rise earlier.
Bad End 3: assuming the Bad End 1 and 2 are averted. Who will Grover call when the di Angelo eventually leave the Hotel? Perhaps no one notice they're slipping out, or the Manticore kidnap them into the Princess Andromeda. A truth about their Mother's Death, a mother they cannot remember, and revelation that they have decades stolen from their life because of gods' action, twisted just right, and the Titan will have TWO children of Big Three working for them. Ones whose Fatal Flaw is Holding Grudge. What about Thalia? Her Fatal Flaw is Power. Luke presents her with the Ophiotaurus, telling her it can give her the power she needs to make her life better. To be free of constant fear of looking behind her shoulders and fearing for her life, especially when her birthday is nearing. And no one can be certain that the Lightning Bolt that almost murder her is not from Zeus. Feeding to her resentment to the gods and make her switching to the Titan's side. The gods are having the Children of Big Three against them, and Artemis will be pinned under the Sky with no one to rescue her.
Bad End 4: Battle of the Labyrinth going better and worse. Better because no one release Typhoon. Worse because no one convince Briares to taking arms and help the camp. No one can figure out how to beat Antaeus. The Camp fell. And Olympus is next. They may not even last till next summer.
Bad End 5: Bad Choices upon Bad Choices. Promotheus trick with the Spirit of Hope? Who will he given it to? Can they be trusted to not fell into temptation and release it? Who will convince the River Gods to not siding against them? Ten Olympians against the Titan's Army. They cannot count for Poseidon to be of help, because he has his hand full with Oceanos. Hades refuses to help due how they treated him, and he had Persephone and Demeter down in Underworld with him. No one pressed a certain demigod from Titan's Army to change sides at last minute, and no one trusted Luke to make the "correct" decision of self-sacrifice, or they too banking on Luke and neglect the threat that is Kronos.
Hence, why Percy Jackson, or whatever incarnate he found himself in, is necessary for Olympus continuous survival.
He was there. On the right place at the right time. He makes the "good" decisions. It was not necessarily "correct" or "the best for the situation", but it was not "a bad decision". He understands the point of the conflict and make steps to prevent it from happening ever again in the future. It doesn't magically make the problem disappears, but lowering the chance of similar conflict arises in the future. And that, that was what vital to Olympus.
Percy Jackson is Lynchpin that necessary to Olympus and the Gods and the Demigods' wellbeing.
Percy Jackson is the recipient or involved in several Prophecies, that together creates a possibility for the Olympus to still standing by end of the day.
And It would do for the gods to keep him Happy and relatively Amicable to the Gods, as well Do Not Have Any Reason to Turn His Back on Olympus.
"You either Dies a Hero or Live Long Enough to See Yourself Become the Villain" should not be applied to Percy Jackson. Because the Olympus is NOT ready to have Percy as their enemy.