My complaints with the World Congress are:
1.) Nobody gets to set an agenda. There is no leader, as there was in Civ V, so the whole things feels meandering and purpose-less. Having alternating leaders selecting specific proposals is probably the best implementation, but barring that, I'd probably be willing to concede to having an extra round of all civs voting for which proposal(s) to bring to the floor.
2.) Many proposals lack interesting trade-offs. Usually, this is because they target a specific civ, rather than all civs. For example: there's the proposal that lets districts act as a culture bomb for a single civ. Why would you ever pick someone else? I'd much rather that this resolution make one specific district type act as a culture bomb, but it affects all civs.
3.) Congress isn't tied into diplomacy at all, except through the favor currency. You can't negotiate for another civ to vote for a specific proposal, or denounce them because they voted for something that you didn't like (in general, I dislike that you can't pick a reason for denouncing another civ, but that's a whole other rabbit hole of a topic). Some back-door politics would help add some life and intrigue into the system.
4.) Diplomatic victory points are back-loaded. One of the problems with Civ V and VI in general is that the steps to any given victory tend to be very back-loaded. Unless you're playing at the highest difficulty settings, you can kind of just grow your empire and expand organically until you have to settle on one victory to pursue over the others (which for me usually happens around Industrial era). There just aren't enough opportunities in GS to earn diplo victory points earlier, which means that early diplo favor isn't as critical as it maybe should be.
I do want to say some positive things as well:
Tying the emergencies into the World Congress was a great idea! Being able to react immediately to a crisis situation is about the only time that the WC ever feels like it's doing what it's designed to do. This is a mechanic that I think I would definitely want to keep if the WC were to be radically re-designed. I also like the Diplomatic Favor, and the fact that it comes from multiple sources other than just city state alliances. Building coalitions gives you clout in the congress, and that's a good design. But I feel like these are the only areas in which this iteration of WC surpasses Civ V's.
I agree with it all!
But I thought in something different for the agenda problem...
The easiest way to designate host/leader would probably just be Diplomatic Favor. Whoever has the most accrued favor when the Congress session starts gets to select which resolution(s) to propose. Maybe the civ with second-most favor picks the second proposal? The A.I.s would probably need to be programmed to not trade away favor as easily, especially as a Congress session approaches.
Other than that (if Firaxis doesn't want to just repeat Civ V's mechanics again), there could maybe be an extra round before the actual Congress in which a leader is chosen. Each civ votes for another civ to be host/leader, but you can't vote for yourself. Top vote-getter picks first resolution, and maybe gets a discount towards the cost of votes, the ability to veto a resolution proposed by another player, or some other benefit? [I'm just spit-balling ideas here...] Runner-up picks the second resolution. We could also use favor to vote for leader/host, but it should probably be refunded for the actual Congress votes.
Being able to negotiate for other civs to vote for you as the host would also be a cool mechanic, especially if there were some way to make promises to them that you'd propose things that would benefit them. Maybe you get grievances if you propose something they oppose? The game would have to make it very clear in the UI which civs favor / oppose which resolutions.
When you put the leader ability on the strongest diplomatic player you have a mechanic that reinforces the snowbally nature of the game and I think it's not very interesting.
I like your second idea, however, where everyone votes to choose a leader for that session.
But, I thought that if we would have another voting session, it would be more or less like this:
First, all nations pour their diplomatic favor on resolutions of their preference according to their interests like it is today but they could select any resolution that is available for that specific era (including competitions maybe).
In the end of this first round, the 2 or 3 most voted resolutions are up to the actual session. This would be defined by the total votes counting both the positive and negative sides of each.
Who voted on elected resolutions preserve their votes and don't have any favor back. Who voted on other resolutions get all favor back.
At the second round, all nations votes for the elected resolutions like we already do today. The civs that voted the first round cannot take back their votes but could spend more favor to reinforce their choices if they have received some favor back from another "lost" resolution.
From now on, everything goes as it is.
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For example, in a scenario where there are only 3 civs and only one resolution can pass, just for ilustrative purposes:
First round, all resolutions of that era are available:
1) Civ A votes 3 times for +100% production cost for units. Civ B votes 2 times for -50% faith cost for units. And Civ C votes 4 times for enhancing amenities from duplicates of cotton.
2) Civ A and Civ B both voted for the same resolution in different sides, but it pass, because it received 5 votes total.
3) Civ A and Civ B get their votes preserved for next round and Civ C get its favor back.
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Second round, only the elected resolution (the mercenary one) is available:
4) Civ A have no more favor to spend, but have their 3 votes for 100% production cost preserved.
5) Civ B decides to spend 50 more favor to put 2 more votes on -50% faith cost for units and now it has 4 votes total from the same civ but from different moments.
6) Civ C decides to vote their 4 times on 100% gold cost.
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Results are decided just like it is now:
7) The +100% cost side was chosen by A (3 times) and C (4 times) = 7 votes.
8) The -50% cost side was chosen only by B (4 times total) = 4 votes. The other side wins.
9) Civ A voted for production 3 times but civ C voted for gold 4 times, so gold wins and the resolution "+100% gold cost for units" is enacted for the next 30 turns.
Maybe it's a bit overcomplicated but it permits a lot more agency for the players with no much harming to the new mechanic logic chosen by firaxis for civ 6 World Congress..