People who complain about this are mostly just being preciously prescriptivist, and they typically do a shockingly poor job of explaining the issue.

To "beg the question", canonically, is to argue in a way that only makes sense if whatever you're arguing to support is true. It's a difficult sort of fallacy to provide examples for because it's quite rare, so most examples you see tend to be constructed for the sake of complaining about the supposed misuse of the phrase "beg the question", and so either trivial or incomprehensible or both.

Too, the phrase "beg the question" itself relies on a rather outmoded sense of the word "beg", whose rarity in modern usage makes the prescribed meaning of the phrase very difficult to intuit compared to the supposedly incorrect one.

All of which begs the question: why should anyone care what complaints linguistic prescriptivists make in the case of this phrase, any more than in the case of any other?

Someone mistranslated Latin into English 300 years ago and now some nerds are mad that people are misinterpreting the mistranslation. Best to just ignore the whole thing by never writing "begging the question." Always write either "raising the question" or "circular reasoning" depending on what you mean.

It's somewhat like the phrase "it beggars belief" - it means that 'belief' is left in a very poor condition, begging alms ...

Don't sweat it, it's a common error made by native speakers too, myself included until I learned about it. :)

It's really not an error anymore because it's been in common usage for years.

Let them be. If we didn't have prescriptivists, everything would literally mean nothing.

And if that was the common usage that people agreed on and understood, that would be perfectly fine. It's literally how language works!