This debate begs the question, is this just another Americanism, like "I could care less"?The term "begs belief" refers to something that is too extraordinary to be believed. It can also refer to something that is not worthy of belief or trust.
I’m not saying the use of the term “beggars belief” is wrong. I’m saying “begs belief” isn’t wrong, as used.
But something that begs belief would mean something likely to make you believe, whereas something that beggars belief would mean something that you find it hard to believe.
(And I'm an etymologist. Well, English Language GCE grade 1.... when grade 1 was good)