I also never recommend to shut down anything, system at idle state draws very little power, just unplug using smart plugs whatever system u don’t want it functional like a heater or whatever, I do that when traveling for a long time, I just hit Travel custom button and it do the rest.
An HA system is there for security, u program it when bad things happen, it alerts you to take actions, this is not recommended at all, I urge u to invest more in securing your home while away!
Answering your quoted question, wireless Thermostat will continue to function without controller, it will stop when batteries !