Up until that point, White had played the whole second half outside of a two minute break at the end of the third. If you want him in at the start of the fourth to buy time without DeRozan (AKA staggering), you're going to have to live with a two minute break at around the five minute mark.
When was the last time Pop ended a game with Mills over White? It might have happened once in the last 50 or 60 games, tbh.
I understand being paranoid when Mills is involved but this is an unfounded worry, IMO.