The thing about Mudd as seen in TOS is that despite the wacky hijinks, there is
an undercurrent of him actually being quite a nasty piece of work who simply disguises his true malevolent nature beneath a veneer of rougish charm. He's the kind of guy who buys you a drink at the bar, and is raising a toast to you with one hand while he sharpens a knife with the other. A reinterpreted version of the character could have a lot of room to play with there.
I do agree that he's hardly the most scintilating of adversaries for a movie adventure that is being aimed at a broad audience. But nevertheless it interests me that the producers of both films had enough of a residual like for the character that they seriously did want to try and give him a cameo in the first movie, and ended up name-checking him in the second.