I don't think it's a logic hole at all, actually. Marty travels back to 1955 and it takes a couple weeks for the changes to catch up to him. If it were different, he would have vanished the moment he pushed his dad, dad, daddio out of the way of old man Baine's car.

The changes play out in real time.

Old Biff gives the book to young Biff, but goes back to his time before the young Biff does anything with it, before the ripple catches up to him.

And yes, there is the deleted scene that seems to fix this.