ZigBee button, double sided scotch tape, Zigbee dongle, HomeAssistant on RPi with a speaker.

That is exactly over engineered crap I want to avoid. I don't want my doorbell to depend on networking or a full blown Linux stack.

To be fair, if you already have a Zigbee network, this is a rather cheap and elegant solution.

Two wires, a button, a bell, and a battery pack.

Yeah this would have been my answer, except for one catch: what do I do with the existing integrated button on the Blink doorbell?

I presume you want to keep the Blink doorbell for the camera functionality?

Then I guess 3d print a casing that just puts the ZigBee button on top of the integrated button?