It could work. How well depends on the fiction, the player, and you.
I appreciate the thing you're trying to do, and yes, Fate is pretty good at handling these kinds of Aspects. However, in cases like this, you'll want to make sure that everyone agrees what they mean.
Take "Fastest Man Alive," for example. In a real-world kind of game, that might mean you can run a mile in less than four minutes. In a superhero game, you might be able to cross a football field in a blink of an eye, but still be beaten by someone who's not a man and even faster. You'll want to have a common understanding of what the player wants the Aspect to do.
Then, moving on from there, Aspects shouldn't replace things like Stunts or Skills, they should complement them. Your "Lovable Billionaire" probably shouldn't have to worry about petty cash expenses, but really big purchases would require moving assets around and making sure that money isn't lost -- and that kind of thing requires a Resources roll. (See also "Fastest Man Alive" and Athletics.)
As an example, if he's a billionaire; buying a new car or a plane is probably not even a challenge for him. Buying a house, or a crate of guns, or uranium? Those could be challenges for anyone, billionaire or not. He'd have to use his Resources skill, and then, if he didn't like the outcome, he could spend the Fate Point and remind you that he's "a lovable billionaire" and get the +2 or the reroll. Bruce Wayne is a Lovable Billionaire; so is Oliver Queen, so is Arthur the drunk. Each of them uses their money differently and at different levels of skill. And if you disagree with where I put the benchmarks, that's fine; you're the GM and you should set them where you feel comfortable. What the aspect does is give you a push when the chips are down and having this characterization can mean the difference between success and failure.
Lastly, remember that an Aspect should always have the capacity to be compelled in ways that cause trouble. Luke Cage, for example, has "Impervious Skin," and that's come back to bite him in one case where his wounds were internal and surgeons couldn't cut into him to perform medicine. There are many ways to hurt someone that don't involve breaking the skin, and so "immune to all damage" doesn't seem like it would be true here. If the intent is immune to all damage, you'd want to add more to it before it becomes a good aspect, possibly with some Stunts as ballast. (And the question of how to handle immunity as an Aspect has been very well answered here.)