Idea, maybe could be implemented in a future update: Replace the Minigame cycle with a feature that lets someone pick one of the minigames. When they do, server-goers have 10-15 seconds to join in. once the time is up, someone else is free to pick a minigame.
Additionally, it could be worth setting up a screen or two in Heart of the Galaxy telling players if a minigame is in waiting, and how much time they have to join. In fact, using the "return home" feature, you could set them up in a few big-traffic spots in the worlds. Like the VSS-Tums Stairwell, or the side of a building in Hi-Oxidar, or even a few set up as corner signs in Outer Wildwood's main boardwalk. (if this is the case, roughly 30 seconds would probably be better.)
All of that said, I hope this doesn't come across as "Make gaem better blease! Make schut!" or anything. It's just a cool Idea I had, I wouldn't want to give any devs more work to do if they didn't want anymore. Game development isn't easy, no matter the game. It's a really great game as is regardless. Thanks for taking the time to make it.