Why is my control panel for AC/Furnace not working when the fuse is ok?
Wanted to follow up in case this helps anyone else. I tested the terminals as Ed Beal mentioned in the comments and was pretty sure the transformer was not working. Since I was uneasy with replacing it myself I decided to call an HVAC professional. Turns out that was a smart idea.

He tested the transformer and confirmed it needed replacing: 120v in, nothing coming out. In the process of replacing it he noticed (and I had totally missed it!) that the power cord from the outlet to the furnace had a significant problem:



So after 20 years with just a bit too much tension on the power cord up in a very warm attic space, it sparked, blew the transformer, and fortunately did not burn the house down.

I have a feeling my installation of the smart thermostat and accidentally blowing the 3A fuse a week earlier was not a pure coincidence and probably made the wiring spark sooner. Me removing the furnace cover and plugging/unplugging a cord that had too much tension on it was probably part of the problem, too. Still, it was an accident waiting to happen.

Anyway, I had the wiring professionally repaired, the 24v transformer has been replaced, and everything is working ok now. Thanks for the pointers, Ed.