I didn't see a calendar hanging up in engineering while it was being bombarded with Delta Radiation. Pike doesn't know the (star)date of the accident.
I don't think he needs to. He can live his life, and see that this is something post Enterprise. He sees there are cadets involved. He would certainly be able to figure out the room when he sees it, so it really is a matter of taking every precaution to prevent an accident involving cadets.
I think it makes total sense. Time travel stories are often about the impossibility of changing things once they are known to happen. Taking the crystal evidently sets Pike on a path where he gets maimed.
Pike's accident was just that--an accident. A terrible situation where he heroically saved the lives of a lot of people at great personal cost. Many accidents are pointless. This one certainly fits in that category. If you know that something is going to malfunction, or there will be an explosion, you make it a point to take steps to prevent that accident. Let's say you are going to slip in the shower. Usually that's because you are not careful. So you BE extra careful, and not slip.
Pike also now knows no harm will ever come to him until that accident, meaning that he knows he will survive. That could make him take extra risks.
Pike ending up in the chair isn't about what day of the week he chooses to go in to work. He ends up in the chair because of the man he chooses to be no matter the consequences.
He can be a hero and avoid the accident.
What made no sense is that he was told that he could walk away from this fate, but can't once he takes the time crystal. What changes by accepting the crystal unless he forgets what he saw? Yet, it doesn't look like he forgot.