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.