No, we didn’t have that problem. We can run the fan independently. In fact, I have an IFTTT recipe that turns on my fan when my indoor Netatmo weather stations detect carbon dioxide over 1500 ppm.
The only time I’ve had issues is dropped wireless, before I bought the OnHub router from Google. The Nest thermostat seems to like it much better than my older router. The only other problem I’ve had is the one everyone had with software version 5.1.3, draining the battery. I just noticed my version is now 5.1.6, so they upgraded in last few days (checked, yup, on January 11). It will, hopefully fix the awful bug that killed my thermostat.
The issue with the battery is not having a C wire, nothing to do with G wire. You’re not using an add-a-wire are you? I believe this works with the G wire.
Anyway, sounds like having the HVAC person out is best option.