People don't normally use the word "beg" in that sense. "I'm begging you, please shut up" is the closest to an ordinary usage of the word -- and even that's being a little melodramatic. If they're talking about highlighting something for discussion, they'd say "you make a good point," or "you raise another issue," or "that brings up a question," or "which leads to another thing," etc.

This leaves "begs the question" as a weird, pretentious-sounding, overly emphatic expression that automatically annoys some percentage of the audience and will slightly confuse the rest (for at least the next decade or so until the purists give up as they did with "irregardless," "could care less," and "literally").

Of course, if your goal truly is to unpredictably annoy semi-arbitrary subsets of people with something as neutral as your speech, then by all means, beg away. Personally, I don't have a choice. I'm a parent, and as is the case with any parent, my kids internalize everything I do. So I can't have fun adding auto-troll expressions into my communication, because they'll do the same without realizing the consequences. I don't like that the world is full of people who both (1) have the power to decide my fate, and (2) set the bozo bit from small things like the quality of my communication, but I'm pragmatic enough to accept that those people do exist and will judge my kids' fate. That's why I speak as if it matters.