[Gogar]

ecobee doorbell will also allow you to use your existing chime,

You could "MAYBE" use the ecobee 'chime adapter' with your wyze and it would be fine. maybe not.

The old doorbell button completes the circuit and sends the power to the bell.

New doorbell cameras are just using that low voltage AC power to run their camera and wifi stuff and the circuit is always on in theory so if you loop your chime in there your old chime will stay pinned on "DING" of the ding-dong.