@mgnl68
It sounds like the internal battery (yes, it has one) on your 1st gen Google Nest Hello Doorbell is failing. The purpose of the internal battery is to provide power to the doorbell during the instant when the doorbell button is pressed and power must be diverted to ring the indoor doorbell chime.
This has now happened to all 3 of our Nest Hello Doorbells, and to hundreds of other customers; see this locked thread with over 1,000 replies:
https://www.googlenestcommunity.com/t5/Cameras-and-Doorbells/Nest-Hello-Doorbell-cuts-out-when-butto...
If you're past the 1-year warranty period, Google Nest says they "cannot" replace the doorbell. However, there is a workaround. If you turn off the "Indoor chime" option for your doorbell, it will stay online when the doorbell button is pressed. And you can use the "Visitor announcements" option (https://support.google.com/googlenest/answer/7672451) to get a doorbell sound and an announcement on your Nest Hubs and speakers. That's what we've done.
Some customers have tried replacing the internal battery, which is difficult because it is not designed to be replaceable. See this iFixit post:
https://www.ifixit.com/Guide/Nest+Hello+Battery+Replacement/131729
