So are we attempting to beg the tone of the board now?
I’m sorry but I don’t think you’re grasping what erislover is saying. His point is not about bended knees-type begging at all.
Instead, think of, for example, the olde English usage, something like “I beg your Honour for leave to Appeal” - ‘beg’ in this context means asking for / requesting something. Now try this again:
“A circular argument is only a special case of the begged question. A begged question is one in which, in the general case, the conclusion is at least as questionable as the assumptions. It quite literally begs the question.”
- In other words, the initial premise is based on an unproven assumption – what is being begged (asked, requested) is that the initial assumption be proven before moving on to accept the veracity of the statement based on that unproven assumption.
Another way to look at it: From the link (above):
“The original sense is of a logical fallacy, of taking for granted or assuming the thing that you are setting out to prove. To take an example, you might say that lying is wrong because we ought always to tell the truth. That’s a circular argument and makes no sense. Another instance is to argue that democracy must be the best form of government because the majority is always right. The fallacy was described by Aristotle in his book on logic in about 350BC. His Greek name for it was turned into Latin as petitio principii and then into English in 1581 as beg the question. Most of our problems arise because the person who translated it made a hash of it. The Latin might better be translated as “laying claim to the principle”.”
- in other words, it’s easier to grasp if you forget ‘begging’ and replace it with ‘laying claim to the principle’. Any clearer ?
Yup, this usage is incorrect according to the “classical” meaning of the phrase, although it’s right up there with the modish meaning complained about by the OP.
A classical piece of question-begging would be “No American ought to do {X}”. “Why not?” “Because it is un-American!”… the “because” merely reiterates the point already made in a different form of words.
Also incorrect 
is your phrasing “you cannot help but ask the question”. It ought to be “you cannot but ask” (i.e. “you are unable to do other than ask”) or “you cannot help asking”. “Cannot help but” is kind of a double negative, though it’s used as if it weren’t. Loads of people do it.
I nominate “to bugger the question.”
Get over it. This phrase is gone. We can’t save it. “Beg the question” has taken on a new meaning in colloquial speech and has gone the way of many other words and phrases which have shifted meaning over time. Big fucking deal. It irks me a bit, too, but, ya know what? I got over it. There’s no sense into keeping its historic meaning, when almost nobody in the general population uses it that way anymore.