as someone who predates fox news, it was started in 1996, by murdoch and ailes, seeking a conservative audience. when you tout carson and leno, carson predates fox news, and leno had the desk 4 years before fox news even started. they weren't even having to compete with conservative talk shows for ratings.
it's funny to see the hit list of guys like colbert and stewart, because they stepped into a vacuum of programming for left leaning hosts. you had fox spewing right wing stuff. you had leno being tame. so daily show and colbert became the interesting thing.
and then the networks chased that youthful audience.
you'll notice the hit lists extend well beyond the networks to people on cable. at which point we are on past affirmative action for conservative talk, or "equal time" -- which reagan gutted btw -- to hunting liberal talk shows where-ever they air, even on liberal cable channels. do i get to literally tell you fox news can't run right wing nuts? that they need to be "fair and balanced," to remind you of a long defunct motto?
we can debate whether a late night show should be conservative or moderate or whatever, but that reeks of the sort of affirmative action/DEI you decry on the left. and you're ignoring the market forces where they are competing with gutman or whatever on the right. if they hire a right winger to host a channel's late night, is that just symbolic and everyone watches fox still. in which case, culture war and institutional seizure as opposed to really wanting to watch network late night for conservatives.