UniFi doorbell requires a UniFi NVR, and maybe UniFi console/ AP. The non-pro doorbells require wifi. POE is only an option on the with the Pro doorbells via a proprietary USB-C to POE cable.
I am currently using 2 of the G4 Pro doorbells. They are direct PoE to my UDM-Pro SE without any adapters. The recordings are stored locally and are private.
Your UDM-Pro-SE is where you run your Unifi Protect. I.e. you need to have another device from the same vendor that supports NVR functionality.
The above mentioned Reolink doorbell will happily run with any NVR from any other vendor that supports relevant standards (ONVIF). Or even without NVR (it has microsd slot for local storage on device).
My UDM-Pro-SE has a 12TB hard drive in it and acts as the NVR itself without issues. I do not need any APs (doorbells are direct wired PoE), and I do not need a separate NVR.
Yes but the UDM-Pro-SE is a separate product from the doorbell. The Reolink doorbell does not need a separate product to function. You can just get a POE injector or POE switch from any brand, run Home Assistant on a Raspberry Pi or NUC and it'll work. The UDM acts as an NVR for your doorbell camera, you couldn't use it without it, right?
You can run UniFi console as software on any* device to manage the doorbell, which would then allow you to use Scrypted as the NVR and expose it to Home Assistant.