I've integrated a Nest E thermostat myself so I know Nest devices work. The setup is a bit of a ball ache though.

For an age I had issues with Google Assistant integrated products though. It would be connected and then lose the connection and sometimes it would just not want to reconnect, for days at a time.

I've found it was mainly due to how I was remote accessing my instance. I was using DuckDNS and Nginx. I moved to using Cloudflare and bought my own domain name, and suddenly everything worked much better.

I would however have Nest drop off and require reauthenicating every time my HA instance was rebooted. This was until I had someone suggest I make my Google Project "Production" rather than "Testing" which stopped that.

I now have a rock solid connection to Google and my Nest thermostat works as expected except for one little quirk where temperatures are displayed in HA 0.2 degrees higher than they actually are.