The transitive use "begs the question X" (where X is some actual question) is commonly used that way and well-understood (and this usage is, while idiomatic in structure, fairly obvious from the usual definitions of the words involved.)
This is readily distinguished from the intransitive use "begging the question" or "begs the question" to refer to the petitio principii fallacy (an idiom based in what was, at the time, a poor translation into English that hasn't gotten better with age.)
Even better, the intransitive use can be seen as (though this reverses the etymology) a special case of the transitive use in which the question begged is the original one the argument sought to address -- the justification for the conclusion which has been assumed in the petitio principii fallacy -- so that the transitive use that you complain about both generalizes the special intransitive case and rationalizes it with common usage.