Having had no luck getting GCM to work on my Lumia 650, I decided to try it on my Windows 10 Pro HTPC, which has a Bluetooth 4.0 LE adapter in the form of an Intel Centrino Advanced-N 6235. Yes, GCM is available for PCs in the Microsoft Store.
It works! Not as convenient as GCM on a smart phone, but more convenient than using a USB cable, mostly.
I can do everything on the HTPC that I would expect to be able to do on my mobile phone. Restarts of GCM cause no problems. Bluetooth seems to stay connected all the time, but if I move out of range of the HTPC Bluetooth connection, my Garmin device (a VVAHR) disconnects and then reconnects as required, and GCM works whether it is already open when connection is lost and reinstated, or if I have closed it and restarted it after moving back into range.
In fact the Bluetooth connection works as I expected it to work on the Lumia 650; the connection is established as required when the two devices are turned on and in range. GCM doesn't need to be running for this to work, which is what I expected, and as Bluetooth is designed.
Why can't Garmin get GCM to work on a W10M device, if it works so well on a W10 PC?