You mentioned needing to replace your primary Wifi router. Assuming you have other Wifi points, did you also factory reset those and set them up again because that's required if you replace your primary router. The easiest way to check if everything is good from a Wifi perspective is to do a mesh test.
Now to your doorbell. If it's 4 years old, you have the original Nest Doorbell (wired) also known as Nest Hello. You mentioned having factory reset it, was that by following these steps? Scroll down to the factory reset section and refer to the steps for Nest Doorbell (wired).
What colour is the light on your doorbell? You can check what the colours mean in this help article. Refer to the Google Nest Doorbell (wired) section. The light ring should be pulsing blue, which means it's ready for setup in the Nest app. It sounds as though you're checking in the Google Home app, but that model of doorbell is setup and managed from a connectivity perspective in the Nest app so it's best to head to that app. Once you get it sorted in the Nest app, it will also be accessible in the Google Home app.
