In my experience the most popular and fun “party games” are boardgames such as Top Ten, Time’s Up, Hot & Cold or Codenames (more or less in that order). They work best for 6 to 10 players. Though I don’t think they shine in a highly competitive tournament setting.
Randomness exists in all of these games but I consider it very balanced/smoothed out so it shouldn’t really affect the outcome. Not all of the games I mentioned have permanent teams, but that can easily be changed with house rules.
Reminds me of Magic Maze, a boardgame in which player play a Barbarian, a Dwarf, an Elf and a Magician who lost all of their gear in battle and now resort to robbing a shopping mall so they can go on their next adventure.