Better yet, keep the vote system in place, but exclude the game mode that was most recently played.
For instance, the vote may be between Elimination, Uplink & Extraction.
Everyone votes Elimination, it’s played, game moves on.
Next vote excludes Elimination, and the choices are now between Uplink, Extraction & Satotage.
Everyone votes Uplink, it’s played, next vote excludes Uplink and Elimination is chooseable again.