There's plenty of American comedy that doesn't fit this mold (Broad City and Search Party are two examples that come to mind).

Definitely. It's rather that the plucky underdog surviving on his wits is practically absent in British comedy.

Brits don't do Bugs Bunny, only Wile E. Coyote.

I think there are also plenty of counterexamples in that direction. Isn’t Edmund Blackadder a plucky underdog surviving on his wits?