You didn’t mention what was going on behind the threesome that was behind you. If the other groups were right behind the threesome, letting the threesome play through is just going to make things worse, as you’ll likely still be waiting to hit on the tee box as the next group is walking up. Even bigger log jam.

If it was wide open behind the threesome and they are waiting on every shot, then sure, let them play through if you like. However, a threesome (or twosome or a single) is going to have to expect to wait if things are crowded. There is no way a foursome can be expected to play faster than a threesome, all other things being equal.