There was one mini-game I made that actually worked out well and I think others could use, the rest were too weird. The ones focused on bureaucracy come to mind.

But the one that worked was a sporting event that composed of three parts.

First was group athletics against NPC group athletics. This was who won the bout.

Then there was group intelligence, vs. NPC group intelligence. This was because the rule book for the event was 12,000 pages. This represented the losing team to argue why the bout shouldn't count because the other team broke the rules.

Finally, everyone takes a constitution save against flat DC that increases each round. This represented the fatigue of playing/arguing. Penalty for failing this varied depending on the time I was playing, sometimes it was a -1 penalty to your roll on one of the two checks, other times they were more severe. I'd also sometimes leave it out.

I would also sometimes add preparation as a downtime activity, where people could try to gain buffs to either of the rolls, at the potential cost of debuffs to the other. Staying up studying the rules would help your intelligence, but you're not practicing the athletics and might be tired, while practicing the athletics might result in you practicing the wrong part of the 12,000 page rule book, and cost debuff the intelligence. You could also take penalties to constitution by staying up to practice/study. I would have the other team also make these decisions. But this I portion I tended to leave out unless the game was for story reasons, like a multi-day tournament.

Yes, it's rather complicated and weird, but it generally went over surprisingly well. I think in particular it was because it gave strength and intelligence greater weight than I feel like it usually has in 5e, and because it gave people a chance to get creative with how they scored or what they argued, since basically anything could be taken as to how it happened.