I have had it switched on on my watch for over 5 years at this point. I record activities pretty much everyday which over the years has been a combination of both running and cycling. In that time I have probably has around 5 false positives, 3 of which I have notices and cancelled before it has sent messages to my family.
I have had a couple of actual crashes in that time and it picked both of them up fine and sent messages as it should do, they were both on the bike. It has never detected any falls I have had whilst running however.
My false positives have never been in the same place, so I keep it on as the couple of times I have had crashes it has worked as intended and although I was fine and able to walk away from them if I hadn't I believe it could be beneficial still.
Injuries can be depressing as fuck when a lot of your life revolves around being active but at least you are getting some good drugs for free! XD
I usually just alternate my training and shift into other things I can still do like for example when I broke my foot running I started doing a lot more cycling as I could do that without pain.
Gotta keep that motivation up as you say though as it can get pretty depressing not being able to do what you love because of body issues, definitely worth while putting in some core work if you can without hurting yourself as core strength is beneficial for everything in your life!