It's sort of a nuanced issue, since a lot of people took The Daily Show seriously in that era anyway. The difference is, of course, that Crossfire and its ilk were/are unserious in subtle ways, and seemed meant to be taken seriously. I'll leave it up to you, the reader, to decide which space Joe Rogan's show lives in (I actually couldn't tell you, never having seen it).
Presumably people watch CNN for different reasons and with different expectations than they watch Comedy Central.
Tucker Carlson's rebuttal was that Jon Stewart could book political guests that wouldn't come on Crossfire, so he had a journalistic duty to ask hard questions.
Which... felt kind of weak to me. For profit media is for profit, and both Crossfire and Daily Show were beholden to the bottom line, and had to calibrate their behavior against that. Edward Roscoe Murrow, they were not.
PS: My favorite Stewart line from that interview remains (paraphrasing) "Crossfire isn't theater? Tucker, you're 35 year old man wearing a bowtie. This is theater."
There's a play by play of it on Crossfire's Wikipedia page also: https://en.wikipedia.org/wiki/Crossfire_(American_TV_program...
https://www.npr.org/2005/01/11/4279560/tipper-gore-and-famil...
I'd also say the same thing about Al Franken - he was pretty deliberate about "de-comedying" or whatever it was he called it when he went into his first term in the Senate. I will admit that that would not have been as much work for him as it would be for Stewart.