Pixel 4a supports bluetooth covid notifications. It took a couple of minutes of it messing about to get working. There's also a covid menu in the phone where it can be enabled / disabled.
I have a notification from NZ Covid Tracer saying "checking exposure keys" with a moving bar. I wonder if it's going to keep doing that, chewing up battery, or if it's just getting itself sorted. I have that app set to not use battery in the background, but I guess I'll have to whitelist it if I want this to work - which I'm not sure is worth the bother in level 1.
The framework is built into the system, it's not part of the app - you don't need it running in the background for this.
Please enable the Bluetooth contact tracing.