On my 20 year old Bryant forced air gas systems, the limit switch cuts the power to the gas solenoid and to the common on the thermostat. Not sure of the logic behind that. The actual furnace logic board stays on and registers an Error 33 (limit switch issue).
I just installed three Ecobee4 thermostats - yes I have three totally separate systems. I have just the thermostat on the system for the master suite, the thermostat and a remote sensor for our upstairs which has a big loft and a big bedroom, and the thermostat down stairs plus two remote sensors for the family room and my office. I’ll see how that goes and add more if needed. I really like the fact that the 4 has a full version of Amazon Alexa built in and has a fairly good speaker. We have a few smart wifi light bulbs that we can turn on and off by talking to the thermostats when we enter rooms which is really nice. I don’t see these thermostats as a way to save energy, in fact I think the process of making specific rooms more comfortable without being able to switch off other rooms will actually use more energy - but comfort is good.