Being a Detroit Metropolitan area resident, I was watching the game and have seen the play you're talking about.
Your question, "... if the batter had not been injured on the play, that he would have been allowed to re-enter the game to finish his at-bat?"
If the batter had not been injured, and based on the original call of HBP, he would have been awarded first base as happened in this particular situation. And, as a result of the review...reversing an HBP to a foul ball, he would have been called back to the plate to finish his AB. There would be no need for a substitution if the batter wasn't injured.