My opinions:
First, in 1993, it would take an extremely strong, popular personality to compete against the Tonight Show. Hall and Miller certainly weren't doing it, despite both being entertaining personalities. Of course, they were both stuck in syndication, instead of having a network contract.
Really, the only possible person who can compete with the Tonight Show (at that time) is Letterman, and if he's on the Tonight Show, well, there's no chance. Even Leno couldn't compete with that (the only reason he survived, IMO, is because of his show's name and history).
So, Miller and Hall with network contracts stick around for a couple of more years, then they both move on to other projects. They are gone long before to still be around into the Monica Lewinsky scandal breaks out. Various other talk show hosts might come in and have their turn in their timeslots, until the networks return to some other kind of really inexpensive programming for that timeslot.
Letterman polishes his act up a bit when he takes over the Tonight Show, with more money behind him, just as he did when he moved networks OTL. He might get a little complacent without competition, so his show loses some of its wackier edge.
It's possible that butterflies can bring someone else in later who can compete, especially with a complacent Letterman. Perhaps just in time to take over Miller or Hall's slot and slowly build up a following. Jon Stewart, maybe, before he becomes big on cable OTL.
Maybe whoever replaces Letterman after the Tonight Show, if it isn't O'Brien, gathers enough of a following to bring them over to compete with him on ABC or CBS -- and that would make Letterman (who probably saw himself as mentoring the guy) very, very mad, and the game would go up a few notches. You know, that could be Dennis Miller right there...