As someone who used a rack-mount child seat years ago (said child is now in grad school), I agree wholeheartedly that they are not a good idea, and it is a lot better to go with a trailer. For one thing, I think that the minimum age for a rack-mount seat needs to be higher/older, than the minimum age for a trailer, since the child needs to be old enough to not fidget or move around too much, or else they can disrupt the stability of the bike.
I never felt insecure while actually riding, since my weight on the bike was providing stability. But when you are stopped for some reason, and either off the seat, or just putting partial weight on it, then a sudden motion by the child can cause the rear wheel to slip or skip sideways. Plus, what if you hit something, and go down?
I now feel fortunate that I never had any bad incidents.
Actually, this discussion reminds me that both of my kids had fairly bad bike accidents as they grew up. My daughter ran her bike into the back of a parked car (she was distracted by some boys playing basketball), and put both hands through the rear hatch glass, severing tendons in her knuckles in the process. My son had a fairly high-speed crash, causing a bad case of road rash (including on his face). Luckily both were wearing helmets (although my son actually cracked his helmet in two), and they were fine in the end. But there were some anxious moments along the way. (In my son's case, watching the MRI being done on his head. In my daughter's case, holding her hands flat while the doctor sutured her tendons, since there was a shortage of available nurses in the ER.)