Just to add a longer-form explanation to the comments I made above. I think what we're seeing now is not so much an increase in the acceptability of the singular 'they' in the english language as a decrease in the acceptability of the generic 'he.'
We had a minor revolt recently (hey, when you're as old as I am, you can call events from the last couple of centuries "recent" with only a modicum of irony) demanding that firm rules be applied to english usage that had not been in existence previously, along the lines of the ban on split infinitives and the terminal preposition (language rules up with which we should not put). But for the most part writers of the language from Chaucer to Dickenson were quite content to use the singular they, and people spoke it every day ("I'll leave a note for the next occupant telling them what happened before I leave").
But for almost as long as the English language has been around, 'they' has been capable of a singular meaning. Often it was paired with "one" to help establish the singular context, as in "one might think they had strayed from the path when confronted with that sight," but it's been acceptable usage for at least 80% of the timespan that generic 'he' has been. And as for agreeing with the verb, if you don't like 'they is' use 'they are,' after all we say 'you are' all the time without ever getting confused over how many of you there are by the second person singular construction.
The generic 'he' has followed right along and generally been accepted as well, but of late that is changing, and as the living thing it is, the rules of the language will change along with it. As such, the days of the generic 'he' are, I suspect, numbered. It may linger around, but start to feel as stilted as constructions like 'one wonders what to make of this' feel today.
As to whether feeling excluded by the generic 'he' has been the primary cause of the sad deficiency in the numbers of female chessplayers, I doubt there's been a study, and I would probably also doubt the veracity of that statement, as well. As a chess coach I have seen enough bad and exclusionary behavior by males toward females to believe there are far stronger reasons than a simple pronoun for that, but I also think that's beside the point.
The point is communicating effectively. Since both the singular 'they' and the generic 'he' have been acceptable usage for most of the lifespan of English, and one is causing more of a ruckus, and more of a distraction, shouldn't we be using the other, if not out of courtesy then out of a simple desire to communicate efficiently?
I'm not going to grammar-police someone's use of the generic 'he;' they can use it if they want to (see what I did there?). They just shouldn't use some non-rule foisted recently upon English speakers to justify it, and should be aware that a growing number of readers will simply stop listening and turn to the next writer if they do. The point of language isn't to follow rules made up by someone sitting off in an ivory tower somewhere; it's to communicate. And if a speaker's choice of words loses their listeners, they would be well advised to revisit that choice, eh?
(With that the old man steps down, folds up his portable soapbox, sticks it into his coat pocket, and trundles off into the darkness.)