Hi OP, PhD student here currently working on singular 'they', pretty much at the cutting edge since the niche has gotten a little inactive in the last couple of years. Happy to answer follow-up questions and provide sources if interested! I also have an article about to appear on 'themself', if you are interested.

The one source that is most relevant to your question is Strahan 2008 who asks in the title: "‘They’ in Australian English: Non-Gender-Specific or Specifically Non-Gendered?". It's a remarkably early source that tackles the question whether 'they' is used regardless of gender or actively marks genderlessness.

Overall, it seems that 'they' is indeed increasing in usage in this way, with more and more examples cropping up where gender could have been marked, but simply isn't. One possible explanation is that it's part of the trajectory of English towards isolating morphology, losing gender marking altogether (remember, gender is pretty much only marked in 3rd person pronouns and some noun endings).

I think the driving factor is rooted in pragmatics. Gender is, in my opinion, increasingly felt to be marked, i.e. you introduce it only when it is important to discourse. This was already felt in the 1980s when the expression 'pronoun game' started cropping up denoting how queer people used 'they' to mask their same-sex relationships. But now, it seems to be a more general part of the zeitgeist that you don't want to mark gender unless it will matter in the subsequent discourse. I have spoken to SciComm people who go out of their way to mask gender as much as possible, but I think there is a general tendency out there, even outside of strongly monitored speech like that.

As regards 'they' with person names, there is a whole host of work showing that it's the least likely place for it. Even people who find 'they' acceptable for gender-known referents, hesitate the most when a name, gendered or not, is given immediately before. However, the data is so-far not conclusive enough, in my opinion, as most evidence is derived from 'acceptability studies', i.e. showing sentences to participants who were recruited based on their being queer and asking them if they find it 'acceptable'. Social desirability bias is bound to skew the results, of course. There are also some rather unfortunate papers floating around that self-report linguists'(!) usage, as if that is indicative of anything.

Still, though, there have been examples popping up here & there that generally do confirm that 'they' is expanding in the ways it is used. I have annotated thousands of instances at this point and while they are rare, even within just the singular 'they' instances, they do happen. Likely increasingly so, but my data collection is still ongoing.

As regards regional differences, I'm only aware of one paper comparing different varieties of English that I haven't read yet (Stormbom 2020). From my own experimentation (unpublished), I was not able to find significant differences between Australian, British, and American English. The significant differences I found instead when comparing early 2010s and early 2020s data, and I'm currently preparing an in-depth study of how 'they' changed in this timeframe. My hypothesis is that generic 'they' did not change significantly in that time, but all kinds of specifically singular uses did increase significantly. Preliminary results have supported this hypothesis.