It's not even in the realm of hyperbole, is it. It may want to be, but it's not. And unless this is something common to "relatable family diner(s)", singling out Applebees is also a head scratcher. The "joke" here is "fucking liberals". When the "fucking liberal" in this case is just a random ass pull.
It's a combination of odd reaches and odd selection of targets that makes one more say "what?" than anything else.
Kind of like how schools don't actually give students detention for missing class during shootings, but the onion said it total happens because of...
It's the same thing. The Onion does it much better, but it's still satire.
I wouldn't fault someone for enjoying the Onion but not being a fan of the Babylon Bee though. In addition to having worse writers the Bee is also limited to pandering to the worldview of their target audience which doesn't always have a solid basis in reality. It's easy for their jokes to fall completely flat when they're a scathing critique of something imaginary, but if you were a believer in the fantasy you'd probably feel the Bee was much more entertaining.
That said, the Bee does manage to land some solid hits at times and I suspect the cognitive dissonance that can cause may make some people uncomfortable which could also sour their enjoyment a bit.
It's why I contrasted it with the Onion's headline about students fleeing school shootings getting detention. All the links to events are there to be made. The Applebees thing lacks those links. In this case Applebees is standing in for Twitter. Which is just a weird stand in. It would be funny if the headline was that MySpace was proactively banning Greene for blah blah blah. Because MySpace is also a social media company, but completely irrelevant, so it's funny because it's essentially "you can't fire me, I quit".
It's not good comedy, but the basis of a joke is there.