No, it’s “dry contact” output.
First step would be to abandon AC-power and the chime…
If your out of the box is as in thinking outside the box you might consider an ESP-01 relay on the chime and an ESP-12F on the push button (requires a couple 10KΩ resistors). This power supply, 3.3VDC or 5VDC), (which the experts here on HA will find questionable) could be used if you power it with the AC5V-AC30V power already being used with the chime.
You’ll need a programmer to program them with Tasmota
You’re already here on HA. MQTT integration, MQTT broker addon and the Tasmota integration.
You’ll want an enclosure (I design and 3D print mine, but a yogurt container works, too)
OK, maybe a bit much. But the price is good; ~USD$10 plus enclosure.
Tried this, still silence. I think I need to abandon this approach. 
Also the approach from @OldSurferDude is way beyond my skills unfortunately.
Maybe I need to look smart chimes and spend a little more money one getting this damn thing smart.
I really appreciate the help and I have definitely learned very much again.
After all the effort, try to find out why. Instead of chime you could wire some indicator light as output for debugging. 1W 12V light bulb for example.
Good option.
If you want diy at minimum cost, substitute that 12VAC power source with 5VDC wallwart or USB-charger, use Plus Uni powered from that and use 5V buzzer to substitute chime. You likely find all components at home…