In the days following that, I was able to read on the web about other people having similar problems with the glove box of their 1-series bimmers. However, it looked like there are no OEM or after market parts to replace the door latch mechanism. You have to buy the entire box, which is shown as number 2 in this diagram. At around $140 it is not cheap. In addition, if you buy a new box, you need to move the lock cylinder from the old glove box to the new one. That may or may not be a DIY job.
Knowing that I can buy a new box if needed, I decided to first try a DIY surgery to the defective door. In the picture below you can see the problematic left latch. It retracts only half way instead of all the way.
The second picture shows the latch being filed to create a 45 degree wedge on the otherwise flat side. The third picture shows the plastic latch with wedges on both sides.
The DIY surgery seems to work. My glove compartment is now no longer coin-operated. I expect the door to continue working until the latch is totally broken. At that point I need to buy a new glove box, or a new car.