Colbert beats Fallon because Colbert has bite and Fallon is weak as dish water. Fallon was perfect for the Obama era, all soft comedy with very little satirical bite, and along comes Trump and Fallon fell down.
Also, when Trump appeared on Fallon, his interview was ridiculed and slammed by many in the US media for not addressing key issues and for humanising Trump to the Fallon audience.
The Late Night shows have not traditionally been intensely political, though, at least not in the manner that Colbert's show is now.
Letterman and Leno weren't interviewing politicians, advisers and journalists in the same way that Colbert immediately did when he took over the Late Show. Obviously Letterman and Leno would have a few monologue jokes about current affairs, but that was a very different beast to what Jon Stewart and Stephen Colbert developed in the early-00s over on Comedy Central. There should still absolutely be a place for a non-political satire late night show. Whether or not one likes Jimmy Fallon's comedy, he should absolutely be able to do a show that is just silly, funny and absurd without a serious attention towards politics.
I totally agree that the Colbert show has benefitted from the Trump administration, as it gave his Late Show a target and a bite and a USP. Perhaps I'm spectacularly naive, though, but I fully suspect that he as a person would rather Trump lost the presidency and he (Colbert) started losing in the ratings to Fallon, rather than the other way round. In fact, he says as much in this interview! https://www.youtube.com/watch?v=xx51IrK8mnM
Stephen is a very deep, religious man, and it is that part of him which brings his (hatred) I don't like using that word, but it is the best I could think of, it brings it to the fore, when the acts of Trump are shown day after day, especially during this pandemic and social unrest in the US, Stephen can't hide his true feelings.
You should watch Stephen being interviewed by CNN's Anderson Cooper, where he really bares his soul, especially talking about the deaths of his father and two brothers in an airplane crash in the seventies, which has left a scar in him.
As a devout Roman Catholic, Stephen too also has problems with the blatant hypocrisy of Trump - I hope I am not getting into trouble by saying this on here, but it is also this, that has affected his comedy.
Also remember he spent a decade pretending to be a different version of himself on the Colbert Report, where he played Stephen Colbert, a right wing, conservative nutter who hosted his talk show.