Wiring Ring Doorbell to existing wires?

We recently changed the front door and the existing doorbell button broke. We're interested in fitting a Ring or Blink doorbell, with Chime accessory so that a noise would be heard in the hall.

The previous doorbell's wiring is still there and, if they touch, the bell still sounds in the hall. Two questions:
Can we wire the new Ring/Blink to these wires? If so, will the old bell sound when the new unit is activated, thus removing the need for a Chime unit?
TIA

The short answer is yes. More details on the product websites.
I have a blink doorbell wired up that way.

Thanks both. Most useful video, ClubMan. Will check the unit!

i never had any success getting a ring to work with an existing chime personally but the add on units work well.