Not sure if it will speed things up, but after clicking on that "basic" site for MyChart I mentioned above, and entering "Texas", got a list of numerous Health care providers/networks.
I looked and found that you still must have an appointment to get the vaccine, and only the County health department can issue those appointments.
That's a bummer! I did notice that some of those links for Texas were also part of the same network I used, CHI Franciscan. Again, it was smooth for me. And given that we are getting the Moderna vaccine (second shot on Thursday), and they are working on a booster shot for the newer variants, we'll (most likely) get notified through MyChart about its availability.
Have you got the both the Safari and Firefox versions of the extension? Yes, but I rarely use the Firefox one. I forgot I had it there. The window did pop up independently of either browser and threw me a little.
OK, I think we've now reached the end of this tale. In the end, the extension was taking good care of you.