They make 2 versions. A battery operated one, and a hardwired one. The hardwired one is only powered from the transformer and existing doorbell wiring from your house. There is no battery inside.
And to prove my point, you can’t even connect a traditional mechanical chime to the wired version of the Ring. People have tried, but it’s physically impossible because it’s missing the required internal parts. But you can connect a mechanical chime to their battery operated doorbell. And this works for the reason you stated, about circuits being broken.
The wired Ring video doorbell connects wirelessly over the internet to chimes you plug in your house. So when you press the doorbell, it doesn’t complete a circuit. It just sends an alert over the internet to your chimes and smartphone. It’s completely different than a regular doorbell.