Yes I always see scotch doubles played that way. The player making last shot in last rack does not shoot first in next rackWe’ve always played the order should stay the same. If a player makes the 9 ball, his partner breaks the next game. If a player misses the last shot and their opponents run out, his partner gets to shoot first in the next game. Even if an easy 9 ball shot is conceded to an opponent, that still counts as a shot and their partner would have to break the following game. If the match involves multiple sets, then naturally the team would be able to start the new set in whatever order they desire.
The question that has always confused me for scotch doubles 9-ball is if a player pushes out after the break and their opponent passes the shot back to them, does that same player shoot or does his partner shoot?