a) Yes, as I said right at the beginning and have repeated, I agree that the comma could be omitted (depending on the rest of the sentence).
However, this is a matter of style and I believe one is doing learners a disservice by presenting them with a rule of thumb, as OWL does, as if not observing it would lead to incorrect English. Worse, EFL teachers might also be misled and mark it wrong or miscorrect it.
If you look into it more closely, you realize that -- mainly because of the particular meaning of "but" in the original sentence -- it should be re-written.
b) Yes and no.
I disagree with what OWL says.
I agree with the style manual I quoted, that a comma is not "normally" (assuming that means more than 50% of cases) used between the parts of a compound predicate -- or rather a compound predicate of the sort they are obviously thinking of. I do, however, think that is over-simplification (partly because it is only one quote from the book, which elsewhere discusses things like "not one, but two") and is not an excuse for switching the brain off.
So we are left with several questions, for instance:
1) What is their (and our) interpretation of a compound predicate?
They are obviously thinking mainly of a predicate with two finite verbs joined by "and" (and some uses of "but"). However, most people who use the term seem to understand "compound predicate" to mean one with two or more verbs. The rules do not apply at all when there are more than two verbs.
2)What do style guides mean by "not normally"? What do other style guides recommend?
Merriam-Webster's, for instance suggest:
"This is an unworkable plan, and has been from the start" [with comma]
"I try to explain to him what I want him to do, and get nowhere. [with comma]
"The board helps to develop the financing and marketing strategies for new corporate divisions, and issues periodic reports on expenditures, revenues, and personnel appointments. [with comma]
The OUP's New Hart's Rules recommends
"The text should be lively and readable, and have touches of humour." [with comma]
Mind the Stop, an older book on punctuation, says ". . . a comma . . . is usually appropriate before but, seldom before and", and wisely remarks
"It is in fact neither possible nor desirable to lay down any hard and fast rule about this sort of thing." It discusses at least one example of a compound predicate where both versions are possible.
As Mencken is (mis-) quoted as saying "For every complex problem there is an answer that is clear, simple, and wrong."