Yes I do have my little doorbell wired linked to my Ring A19 smart bulb + bridge. It does work, however I find this doorbell’s motion sensory range to be poor, especially at night. Also in general the Ring doorbells and cameras (even spotlight and floodlight cams) will only turn on linked Ring lights for 1 minute, until triggered again by a new motion event. The light can be set in the settings to supposedly stay on for 3 or 5 minutes too, but IT DOES NOT WORK. People have mentioned this for quite some time, but Ring has not fixed it.
The lights activate each other much more smoothly and they operate more as intended. You might consider using one of the motion detecting Ring smart lights as the motion sentinel that triggers your other lights, instead of linking to the doorbell directly.
But yes lights connected to bridge do work with the doorbell. It’s just limited in my experience. The main limitation being the very poor motion detection of the video doorbell wired at night. If on the other hand your plan is to use the Ring lighting to trigger your doorbell to record sooner, that should work well enough.