Outside of Amy Schumer, how popular is stand-up right now?
Amy Schumer is popular for her sketch comedy. At this point, her whole standup career can be seen as a lead-in to her getting a show on CC.
Plus he's eating off all of this too, and I doubt he'd really want to be a household name - his content would be under more scrutiny if he was.
No need to doubt. Burr has repeatedly said he would prefer to remain a touring comic rather than have his own show and be famous. He contrasts himself to Louis CK all the time.
In fact, if "F is for Family" was a real sitcom and not a cartoon, he probably wouldn't have done it.