His story indicated he left voluntarily, because he decided to interfere with the lives of everyone else, instead of sitting on a mountain, watching the world pass by. He was only expelled, because he left the mountains to use his Voice and intervene.

Where did you hear that? I was aware he'd spent time with the graybeards but not that they wanted him to be one

Believe y'all are overthinking this. There's no "thu'um to reverse aging." It just tempers you, like iron into steel.

They did, which is why they were furious when he left High Hrothgar to fight in the war. When one learns Thu'um, they're pretty much contracted to stay in High Hrothgar, as one really isn't use Thu'um outside of it.

Ulfric, however, doesn't give a [censored].

He's far from the first to break the Greybeards' unilateral monopoly on the Voice.

But he's the latest.

He have unique dialoge which is only avaivable after discovering that you are dragonborn and before you go to the Greybeards.

If there was a Shout to reduce aging, the graybeards wouldnt have gray beards. They would all be like 20.

Meditating and focusing on this magic probably just slows down the aging process automatically. Like a good diet but better.

"Speak only in True Need."

I bet there is a Shout to reduce aging. Or it's possible that it could be created. After all, Dragonrend was created by mortals, so Shouts can be created. But even if there is and the Greybeards know it, they don't use it. Part of their philosophy, I'd guess. I'd bet my money on the fact that the Greybeards live isolated away from most people, so they don't fall ill, and all that meditation helps. It wouldn't be farfetched to say that Greybeards live to 120 or so, after all, even Tiber Septim lived to 108 and he knew the Voice.

When you meet Arngeir, doesn't he say he has studied the way of the voice, but he never thought he would ever meet a dragonborn, which means he can't be older than Alduin, or any older than when the last dragonborn walked the earth, they probably live to a nice old age but i'm not sure about hundreds of years if I'm right.

The 1st PGE describes "the most ancient and powerful of the Tongues," who "have spoken once only in living memory."

Apparently our notions of their longevity come from here.

They are old because they are not only living in seclusion at the top of a magic mountain but also because even men of peace can be hard to kill when they can shout you on fire when backed into a corner.

Supernaturally old? Maybe, maybe not. But definitely not the sort to habitually die young.

They are "ancient." You have to put it down to the gaga-for-the-Fatherland author's hyperbole.

The author of the First Pocket Guide usually had no idea what he was talking about. In fact, that seems to be the point. This is Tiber propaganda, not a benevolent guide to the Empire. These are the villains of Redguard writing this.

They might be old, but even Arngeir isn't old enough to have ever seen a Dragonborn.

Paarthurnax states that the Dov are immortal. so why would a Dov, a creator of the Dragon Shouts, need an age-reversal shout? They're immortal none-the-less. Arngeir also mentions that the 'DragonRend' Shout was the only shout made by mortals. So I don't think there is a age-reversal shout.
But I could be wrong on this statement.