Yeesh, that is one of my pet peeves. It may raise or suggest the question, but it certainly does not beg the question.

Of the 323,000 references to this phrase turned up by google, about 2 or 3 percent are people who have either unilaterally decided or have accepted the wisdom of some other, petitio principii-aware, usage nazi, that the only acceptable usage of this phrase is the classical rhetorical fallacy usage:

To beg the question means 'to assume the truth of the very point being raised in a question'.

The other 90%+, found in many highly respectable sources, including The New York Times, The Wall Street Journal, The Economist, The Times Literary Supplement, and even a hard-core academic journals, are usages similar to mine above, where the verb 'begs' is used as a substitute for the word 'entreat' or the phrase "ask earnestly for or of'.

1. Language is a live, mutating entity and 'new' forms of usage are being adopted all the time.

   The only 'static' languages, are dead languages--like Latin.

2. The classical usage is itself suspect.

   Let's try a little substitution--'beg' for 'assume':

   'to beg the truth of the very point being raised in a question'

   Does that make equivalent sense to the classical definition above? I think not.

   Or 'beg the point in a dispute' as meaning 'To take for granted without proof'?

   However, try:

   'That entreats the question...' or 'That implores the question "...", be asked. or 'That craves the question...'.

   I think those do!?

   Do you think it is possible that some ancient scholar made an error when translating from Latin or Greek to English or French at some point in history, and as a result, that nonsensical, idiomatic phrase has become enshrined in classical rhetorical teaching?

3. Your suggested alternatives--"It may raise or suggest the question,..."--do not capture the essence of this usage.

   The implication of the phrase in the usage is not that the original text raised the question.

   It is that the original text didn't ask the question, when it probably should have asked.

   Whilst that is absolutely different from the classical usage, it does coincide with various other usages of the word 'beg' as a substitute for the word 'ask'.

   As in, 'I beg your forgiveness', or 'I beg to differ', or 'They begged the court's indulgance'.