I’m not sure off the top of my head. There’s a jessie specific hack related to the wifi, but the section about the LCD software looks like it’s all stock sutff.
The docs on touch screen support only seem to talk about the kind that attach via the ribbon cable.
I wouldn’t be surprised if the LCD-show package goes away from raspberrypi.com/Downloads at some point as it appears to be about 5 years old. I found a repo on GitHub that might be that same LCS-show package. That was updated a month ago, so it seems like trying the latest version is a logical first step.
I’ll try to reproduce your results and then see if I can get it working on Buster. Glad to hear that stretch passed a smoke test. Spring and Fall are a great time to take my thermostat offline. 