Yeah before you shared that article, which used the murder/abortion example...
(think it was Ollie who shared that but whatever)
...of a consequent implied within the antecedent, I wasn’t aware of the origin of the phrase - nor yet do I understand how the phrase “begs the question” ever meant what it meant, and not what it is now used to mean. I do find it interesting. If anything, I think it could be an example of a phrase taking on a more intuitive meaning over time. But yes I also agree that in general any term or phrase losing in popular usage its semantic roots is tragic, if even justified like in this case.
I'm guessing that when one says to "beg the question" - beg here means to sort of ask the question for help or support. In other words one is asking the question for help in proving itself, which clearly can't be done.
When I see the phrase used, I find it tends to be by journalists, normally being sarcastic and thinking they're clever in this arch way of speaking "So and so did this which does rather, ahem get me, beg the question of.... " and it annoys me cos it's a kind of misplaced confidence in their own braininess - but maybe that is just me.