Letterman could have taken over depending on if Carson left in the mid-late 1980s/1990s or even in 1992. Letterman was Carson's preferred successor. It'd be interesting to see him compete with Leno getting a talk show on CBS. I don't think Leno would have done well. He wouldn't have the Tonight Show to use, and OTL he was losing to Letterman in the ratings prior to the Hugh Grant interview in 1995.
It's old hat to bash Leno, but he was not a good host. He didn't have good rapport with his guests, he didn't tell good jokes, and with Leno, the Tonight Show lost it's vibe of being an event. Fallon brought that back. Say what you may, but Fallon has been very good at making an interesting show around him if not because of him specifically, and that's what Carson did. Leno had a very lame audience that were more conservative than the Carson audience, which you may think would be old and conservative. Carson's show could get a little raunchy, and that was the fun. You can literally find clips of Rodney Dangerfield and Don Rickles doing the sort of material they did on Carson on Leno, and the audience reacting with this uncomfortableness and sometimes revulsion like they had gone too far (and at worst, not getting the joke). That's how lame Leno's audience was. Just these Middle America lame-os on vacation with a deep existential terror over the fact that they own genitalia.
EDIT:
Leno and his iteration of the Tonight Show honestly feels like a cruise ship comic on some sort of family or 40-something cruise, or maybe a show in Branson, Missouri. He fits the mold, he's just the right niche to be a headliner on something that like, but he's not good enough to have a career elsewhere, and you might as well see him for something to do. And in his own world, he's a major thing. But it's something of what a modern person thinks the old school showbiz was like, and maybe the audience thinks it too, while not actually being what they were like. And that is shown by the example of when Leno brought old school people on his show only for the audience to not get them or not like them. The headliner is a bowdlerized sellout, and the audience is just plain afraid of the world outside their bubble of comfort.