OK, I'll bite.
All of the people arguing for the firing of the training staff: Which specific injuries should they have been able to prevent?
First of all, I'm not advocating that any heads roll, yet. There will be plenty of time for that if the unlikely comes to pass and the team actually does miss the playoffs.
Second, I'm not sure whether "they" should refer to Tito, Curt Young or the MD staff, but "they" missed some obvious signs and pulled Matsuzaka, Buchholz and Bedard too late in three specific games. It was the subject of extended debate in the game threads, with enough supporting dot-chart evidence to make heads spin. If we can see it, "they" should too, or they should hire "us". Each pitcher was pulled at least two batters too late, and in Bedard's case, two innings too late.
Matsuzaka shredded his elbow against Seattle on April 29. It was clear by the fourth (Tek noticed it first), and there is no way he should have been pitching the 5th. Would an earlier hook have prevented the ultimate ligament tear? Maybe, but probably not. Still, the symptoms were recognized late in the dugout.
Buchholz went way overboard against Detroit on May 18. 127 pitches seems too many for Buchholz's rather lean frame, more than he had ever thrown before by a substantial margin, and he was visibly losing both velocity and command on the fastball in the 7th inning. I think that was the game that put the real strain on his back. Should he have been pulled earlier? Yeah, I think so, but he was pitching very well. I put it down to excessive reliance on the 330-pitch rule, on which there has been radio silence ever since.
Bedard injured himself by the 4th inning against the Rangers, but Tito let him pitch through the 6th, commenting during a dugout interview that he'd let Bedard pitch as long as he got outs. The lat injury accumulated during innings 4-5-6. Should Bedard have been pulled in the 4th? Absolutely: it was a no-brainer. Bedard was a playoff investment, and the resource should have been conserved.
I repeat: I do not think heads should roll -- no autopsies, trials or executions until confirmation of death. But there were a few pitcher management decisions that failed. If it comes to blame, I'd be inclined to blame not the manager or the medical staff, but pitching coach Curt Young, who is in the dugout watching every pitch as it is delivered.