I think this touches upon what I will call the whole 'edgelord*' discussion -- when is someone wanting to play something special positive engagement with a creative endeavor and when is it a self-serving attempt to narratively dominate a shared fictional landscape with their own pet project.
*ugh, that word.
In general, I think the best advice/general trend for what works is that you should look for hooks that are the first orange in a narrative landscape, not one that turns over all the apple carts. Much of the above Tolkien examples fall into this category. There are unique monsters, each with a non-repeated schtick. Boromir has a moment of badassery (/preternatural timing) that shows a little of how special they are... but replicates nothing more than having been summoned for the great Council of Rivendell. Even Tom Bombadil's case of being some bizarre little demigod possibly immune to the 'dark lord's plot to conquer the world' narrative everyone else has to go through works because it is specific to his own little hex in the wilderness and can't be used to derail the main plot or premise of the fictional world.
Luke Crane's example of someone wanting to play the last mage is actually one that I would use with the most caution/only with complete buy-in from everyone else involved. Being the last mage in the world has profound impact on what happens for the rest of the world (and thus what the GM can do, and also realistically what they have to do); it impacts what other players/characters can play as, do, and react to the now-unique fellow characters (and also risks them feeling like playing a support role in someone else's story); and it means that the entire game world reacts profoundly to that one character making significant decisions/dying/succeeding/failing.
Again, this is a spectrum, and there are few right answers excepting what does or doesn't feel right. We all (of a certain length of gaming) experienced Drizzt clones, and that was clearly a cliché too far. At the same time, some even more well-worn tropes like small village boy dreaming of growing up to become a (skewed vision of what it means to be a) great knight are effectively ever-fresh.
I think you are seeing the real-concern scenario of a GM-novel on top of this one. To be sure, this kind of situation will have more collective-fiction elements at the expense of shared-game elements than other campaigns, but it's not inherent that this will be a GM-dominated event.