I didn't say anything about playing him injured, it's about signing him knowing he had a back problem. The extent of the injury was the main bone of contention and naturally you'd get an optimistic view on healing time from the player's side because he's keen to join.
The point I'm making is that Wenger ultimately makes the call after listening to medical advice. I have not actually disagreed with you here. But more often than not, his judgement is questionable, especially when it comes to managing the playing side. You also have to look at why we left the signing so late and didn't have credible alternatives.