I maybe wrong but I don't think it's the players name people really want to be free.
I know I have brought up changing a players number to be free a few times and to be honest the only reason I've been going about is because Nordeus said that it was going to happen back in August 2012. Its more the fact that they said they would do a simple change over a year ago and still haven't done it. I agree if changing a players name was free it would end up in chaos.
As for the injuries, I don't have the same problem, im lucky if I get about 7 injuries a season and never more than 1 at a time and I do train my players everyday.