No, the servo can still move. I have used a LOT of servos and when they fail with broken teeth in the geartrain, it can be deceptive. The servo might behave properly until it reaches that one point on the gear where the tooth is broken. It might click a few times skipping over the broken tooth, then grab and move to complete teeth and start rotating normally.
Try this. Run the servo so it moves continuously. Then apply some friction to the output shaft. Not hard enough to stop the servo, but tight enough that you make the motor work a little.
When it gets to the spot where it used to click and then continue rotating, I think that now it will just keep clicking until you let go.