Kanade got Yuzuru's heart. She got stuck because she could not say thank you. (how did she end up in the school BEFORE Yuzuru?)
Yuzuru had already passed on to ........., because his life was fulfilled (no head trauma remember? why amnesia?)
Yuzuru saw that Kanade had gotten stuck because of his heart. He somehow erases his own memory to go to that school.
But there is a bug with the school program. People that enter the program this way only have a slim chance of getting back out. (like the angel player programmer?)
So Yuzuru instinctively gets close to Kanade. his proximity to his heart restores his memories UP TO the point of his death. not his afterlife.
He helps everyone more forward, he finds finally that he cannot move on because of the bug.
Yuzuru hopes Kanade will help and stay, but she is already ready to move on. (This was pretty much the ULTIMATE reason he came to the school!)
So now he is stuck, and there is she can do but WAIT for that slim chance that he gets out.
Some unspecified time later, he succeeds, and together they move into their next life.

Otonashi actually already crossed over. not yet reborn. which is why he had no memories. but Tachibana got stuck so he went back to help.