I used to drop start all my saws, but since I learned the method talked about where you stick it in your thigh, that's how I do it now, except for the climbing saws. Those little pecker pole pruners still get drop started.
It makes sense to me to set the brake, set the throttle/choke control, hit the decompression valve, stick the rear handle under my thigh, and pull. It gives total control, is effortless, safe, once the habit is formed, fast and easy.
I will admit, after drop starting for more than ten years, it took some doing to make proper starting a habit.
As for actual injury, it is hard to quantify, but I started suffering from CTS symptoms, and changed the way I had done things for years to manage it. Starting saws properly was one of those changes.
I suppose wrist elbow and shoulder injuries are also hard to quantify. The safety guys could probably give exact numbers of people that actually cut themselves drop starting, it's like a lot of other things, learning to do it right pays off.