Thanks for the data, that's very interesting. It's easy to see how the meaning changed as the old usage is not at all intuitive to the modern ear.
But as for 'raises the question' I'm sure it is fairly common, I get 23 m hits on nytimes.com. I have no idea where to find a count of usage like you did for 'begs the question'. I also don't really have a problem with using it in general. I just take it to mean bringing a question to someone's attention with no particular urgency (or lack thereof) attached.
Despite my grumpiness about English prescriptivism (and having to wade through the seas of absurd self-reinforced linguistic ignorance that usually comes with it) I don't really have a problem if people use "raise a question" or "beg a question" or "ask a question" or much to the prescriptivist's chagrin "aks a question" since "ask" was actually "aks" at around the same time "begs the question" was translated into English and had been for at least 600 years (from "acsian")and hadn't yet been influenced by the Scandinavian form "aeske".
The most important quality of all in language is the simple question "does it succeed in communicating?" I've seen many writings, in dense, linguistically correct, nuanced English that utterly failed to communicate at all, and many writings by semi-literate teenagers that succeeded despite great irregularities in their usage of the tongue.
Every English speaker knows what "begs the question" means in the modern meaning, it is effective communication. Very few people know in the old. It is not effective to communicate using the old meaning with a modern general audience. That is poor use of Language.
But as for 'raises the question' I'm sure it is fairly common, I get 23 m hits on nytimes.com. I have no idea where to find a count of usage like you did for 'begs the question'. I also don't really have a problem with using it in general. I just take it to mean bringing a question to someone's attention with no particular urgency (or lack thereof) attached.