I have two window units and a switch on the wall wired in to turn power on and off to them like yours. It would be the same as pulling the plug. I added a Shelly relay to each of them to turn them on and off via home assistant.

I can’t change any settings on them as they are physical dials on the front of the unit. However, I generally keep the dials the same and just turn them on and off with the relays.

There’s a warning not to turn the units off and then back on in 3 minutes (I’ve seen this on quite a few units), but I think lots of units only enable the fan if they are switched on in that time frame as a precaution. The compressor comes on after the warning period passes. No idea why.

I use a Broadlink RM4 for exactly this! Works great once set up. Had to let it connect to the internet to set up the first time, then blocked it on my iOT vlan so only Home assistant can talk to it. Has worked without any intervention for several years.