Hi Benedito,
Thanks for letting us know about your experience. We’re sorry to hear that you’re experiencing a battery issue with your Nest thermostat, and us coming short in providing the much needed support.
Wi-Fi disconnection, and power issues may be connected. Nest thermostats are designed to charge their battery with the power your system wires supply. If the system wires can't deliver enough power, the thermostat battery will start draining, and this results in your thermostat limiting its features such as Wi-Fi connection to preserve battery.
You may try to completely disconnect your Nest thermostat from your Wi-Fi network, and use it offline for at least 24 hours to see if you get any battery issue. If the issue gets fixed, the battery drain could be caused by your Wi-Fi network.
An alternative solution to the battery drain may be installing a common(C) wire in your HVAC system. This wire can deliver constant power to keep your thermostat charged up and connected to your Wi-Fi network. We recommend to contact a local Nest pro for further resolution. You can find more details on contacting a local Nest pro here: https://support.google.com/googlenest/answer/9850152
We're always here. If you'd like us to take a look again, please reach out here: https://support.google.com/googlenest/contactflow