Watch vs watches isn’t plural vs singular, just the oddball ending for (most) third person singular verbs. I think “anybody watches” sounds wrong because it has an implicit auxiliary verb, eg, “does anybody watch” (the auxiliary verb gets the third person singular ending).
But, it’s risky to follow grammar advice from a scientist.
Correct-a-mundo as to anyone and anybody. “Any” definitely reflects singular. Everybody? Not so sure.
I’m glas the rest of you are following along with my attempt to scare away the newbies by lettering them think we’re a bit crazy so that I have more time to select my wine.