It also opens the game up in terms of how many puzzles you can be thinking about at once; if you're stuck in a traditional 1-protagonist game you might only have one or two things you think you need to do next, at least in a pretty linear game, but now you always have at least 3 things to work out at once...
[1] I'm not a big IF buff, but I guess Infocom's "Suspended" (1983) is a classic game where you can control different "characters" at once, and I really loved "T-Zero" (1991), where you also open up 3 different timezones you can travel between at will, and changes in the past affect the present/future in a slightly DOTT-esque manner - although it's a very different game.
Suspended: https://en.wikipedia.org/wiki/Suspended_(video_game)
T-Zero: https://ifdb.tads.org/viewgame?id=u8qqrwutdugkexpr or reviewed on filfre.net here: https://www.filfre.net/2017/12/the-text-adventures-of-1991/