diy

I would call it either a control panel or a button panel, but would defer to whatever the owner's manual calls it in a schematic.

bizarrely, those thermostat controllers can be pretty cheap to replace and totally in the realm of DIY.

the wires leading to HVAC should be color coded and sometimes are tagged.

just saying if trying to fix a cheap component becomes too infuriating (because they are often cheaply made).

anyway, good luck

Id just replace the whole unit. Turn off the breaker power, use a touchless tester to verify it’s off, take pictures of the wires and where they’re connected, disconnect.

Edit: I’ve read Ecobee thermostats are good

That's a Honeywell T6. It's really easy to replace, theres a separate part that retains the wires that you can keep attached to the wall. IIRC, it's just one screw on the bottom and then it pops off.

You can find open box/used ones on eBay for less than $50. There's an installer menu on these thermostats, go into that and mark down the settings, and set the new one up the same way. You'll find a PDF of the installers manual online that can tell you how to get into the menu, or I can help you.

You can probably glue it back on with some super glue. Just a little on the edges of the rubbery cover should do it.

If you want something less permanent just slap some clear packing tape over it. You'll still have enough "give" in the tape to push the buttons and since it's clear you can read what's under the tape. Then if you want to try a bigger fix later you can remove the tape.

My plan is to use a cotton swab to very gingerly push the buttons. I assume the panel is less likely to fall off. Why? I only did a few test pushes with a swab but it seems better than a fingertip. The swab focuses the push right on the button and the swab seemed not to be nearly as sticky to the panel as my fingertip.

Why didn't they make it just five fucking buttons? Sigh.