Yes, it will work.
When you say the NEST occasionally cannot be reached by WiFi, that's another matter. For a while I thought it was the WiFi router, that NEST is particular re the brand/model. Accidentally one day I noticed that the iPhone app always connected if I had previously closed the app. Hmm... Sure enough, when I remember to close the app after I have made the change to the thermostat, I connect every time! Leave the app running in the background and it refuses to connect. That annoying blue circling thingy keeps circling then you get a troubleshooting "help" that's useless.
One more thought. Our offices and my home are two floor buildings. Heated air goes up the stairs to the 2nd floor; that's life. The HVACs do not have zone dampers (the office did (Lennox) but we chucked them as there was so much bypass of hi presssure air. Instead we place wood plates over the floor registers on the floor that's not being conditioned (summer = 2nd floor is cooling, winter = 1st floor is heating). In the offices we have had two NEST thermostats in parallel. We use the one we need and turn off the other. In the home we did not do this. So today we are buying the 2nd home thermostat so that when we retire for the night (it's 11F outside now - BRRRR) since heat rises, why control the 1st floor when we are on the 2nd? We have WaterFurnace Ground Source heat pumps plus an oil fired hot water heater and hot water packs in the air handlers heat the buildings in winter as our heat pumps cannot extract enough energy.