No, he doesn't decide whether he plays or not, the manager picks the team based on the information he gathers through training and his medical staff. A club with a proper medical team will tell you if a player is a risk, whether it's Pogba or Rashford, the signs were there to see, in the case of Rashford it was painfully obvious to the point many people were calling this a mistake prior to the double fracture.
At that point, you have to make a decision as a manager to decide the risk. I couldn't care less what a player tells me, how he can play through this, because they all say that, we've all done it. Hindsight is always 20/20, but the more information comes out on both cases, the clearer it becomes that this could have easily been avoided. We're just not going to agree on this if you think a player saying he's fine to play through injuries dictates that he should be to blame if he aggravates his injury, despite the fact that there's a medical staff and a manager that is supposedly looking at the big picture and not immediate results as if the team was in the running for the league.
You won't find a player that won't want to play through injuries, but they're not doctors and they might not realize how compromised they are, it's solely the medical staff and the manager's responsibility to manage that. Pogba can't pick himself on the team, neither can Rashford. So I really can't understand how you think any players should have some blame.