This is a very common complaint about Ring doorbells. They don't play well with mechanical / legacy doorbells. There were some firmware updates that were supposed to help in - have you updated the firmware on your Ring?

I'll second the comment below. It is easy to test if the doorbell itself is working. The typical setup is a 24 VAC transformer with a simple switch as the doorbell. If you remove those two wires from the old switch (that you would connect to the Ring), and touch them together, the doorbell should ring (this is the same thing as using a jumper). If that works, then the problem is with the Ring - either the configuration or the firmware (or bad hardware).