Except that from everything I’ve read about the music industry, the person who gets paid the most out of song royalites is the writer. It’s why Robbie Robertson ended up disliked by the other members of The Band (as he was the only listed “writer”, he got all the royalties), why George Harrison got pissed off at the other Beatles, why “Weird Al” gets permission to re-list himself as the writer of his parodies, etc. Someone doing a cover of a song is giving more money to the original writer than they’re getting for themselves from the cover.
Besides, if a mediocre group covering a good song makes more money than a mediocre group with an original but mediocre song, why isn’t the radio flooded with covers? As it stands, cover songs are pretty rare - maybe one in twenty or thirty songs I hear on the radio is a cover of someone. And maybe one half of those songs are covers of actually popular songs, which it would seem to need to be for it to be an instant cash cow. I’d never heard “Drift Away” before Uncle Cracker did it, and I doubt most people who listen to Gnarles Barkley have clue one who the Violent Femmes are.
I think the other reasons are more likely - it’s a way to get a new audience for an old song that the performer likes.