Detect old doorbell presses and open the door remotely

Hmm not many devices that can detect closure per se. Check out the Monoprice 15260 model, it has an external N/C and N/O contact sensor which potentially can be wired to your door bell. I’m not sure about the voltage requirement. Usually it’s a no voltage scenario but maybe you can wire it to use no
Voltage and then trigger a separate door bell and chime it: [RELEASE] Monoprice Door Window Mailbox Z-Wave Plus Sensor (Recessed/Mounted - 15268/15270/24259) with External Trigger Option and Tamper Device Handler (Official)

For a list of potential relays to trigger secondary devicessee the second post on this topic: [RELEASE] Virtual Garage Door Opener/Controller with Relays and Contact/Tilt/Door Sensors