First, the people as vehicles means it sees a vehicle in the frame when it does the analysis, so may need to make some adjustments to try and reduce that. Detection zone being the most useful.
Technically video doorbells are polarity sensitive, but a lot have built in circuitry to reverse it for you if needed. Not sure if this one does or not, if the instructions say it doesn’t matter, then it probably does have that built in.
To clarify, is your mechanical chime itself buzzing (and are the plunger(s) up against the metal plates as though the button is being held in)? Or is it just the transformer? If it is just the transformer, I would suspect it is just being stressed to its max ability and may need to be upgraded. Many homes have 16v 10va which over time can be even lower, and that is barely enough to run the doorbell. 20v 20va would be a big step up and they’re pretty cheap.
There has been discussion here about a chime controller being needed on some of the doorbells, but I’m not familiar with which cases that is (and seems a bit odd to me that you’d need something to interface your mechanical chime in).
The biggest challenge is old mechanical doorbells and standard buttons were pretty forgiving about wiring and a lot of houses they’re a mess. A second button at the back door adds complexity too (do you still have a jumper/diode on that maybe from your previous install)?
EDIT - take a look in this thread where it seems the chime controller is required if you want to operate your mechanical chime (the symptoms there are a bit different but required is required regardless)