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The nearest to is amenities, and demand for ‘em does not increment in quite such an expansion-limiting way.
Six is back to supporting as much REx as one can manage, more-or-less.
But district cost increases with the more number of techs and civics you research, and builder/settler cost increases with each one you build or capture. So eventually you reach a point where the investment required to get another one out just isn't worth it anymore relative to how many turns you have left to win.
Amenities kind of works similar to happiness, but there are so many sources of them between entertainment districts, policy cards, and great people that it's not really a limiting factor on the number of cities, and more a limit of how many HUGE cities you can manage.
That's not how the AI votes in that resolution. Your relationship with the AI has zero effect on which luxury they will ban. They ban the luxury they don't own that has the highest number of copies in the world. Open the Global Resources Report screen, go to the bottom, see which luxury is the last (the one that has the most copies). Anyone that doesn't own that luxury will vote to ban it. The ones that own it will vote for the next they don't own in that list.
Amenities are almost meaningless in Civ VI not only because the sources of amenities are abundant, but also because of how amenities are consumed by cities. Each city needs one every 2 citizens, starting from the third citizen, so your city needs one when it has 3 citizens, then 5, then 7 and so on. In early game it's specially easy to keep your cities happy, since they will only start consuming amenities once they get to 3 pop.
The bonuses and penalties from amenities aren't as crippling as happiness in Civ V, so even if you end up with negative amenities in some of your cities, the impact is negligible and easily surpassed by the advantages of having as much cities as you can manage to settle. Players usually sell their luxuries in early game, because it really isn't necessary.
Amenities can only become an issue in the context of loyalty. If you're having some serious loyalty issues, the difference between having +3 amenities and -3 amenities in a city is 12 loyalty. Having surplus amenities can be really helpful and having negative amenities can hurt you quite a lot.
I didn't say lack of amenities was crippling, just significant enough to moderate my expansion. From what I understand, your amenities are spread out over your empire. So e.g., if you have 4 natural amenities and 4 cities, it's 1 to each. But if it's 4 natural amenities to 8 cities, of which some have access to excess amenities from other sources relative to other cities, the excess natural amenities will be distributed first in priority to cities that need them most and not stack with your other amenity sources in cities with an excess of them. So it does have a significant effect on growth if you expand too quickly before establishing excess or what I'd call "buffer" amenities.
It certainly matches my experience of votes for the LP resolution—though I didn’t know the exact rules, just noting how often coastal luxuries are the target of the vote is a solid demonstration. Not an “official source,” of course.
https://civilization.fandom.com/wiki/World_Congress_(Civ6)
It doesn't, because each city you settle will need to grow to 3 citizens before it even needs one amenity, then to 5 before it needs 2. 8 cities with 4 pop each requires only 2 luxuries to stay content and there isn't much need to keep them content. A displeased city (-1 to -2 amenities) get only a 5% penalty to yields, which is negligible at this stage, mostly compared to the advantages of having more cities. With 2 luxuries, you can grow 8 cities up to 8 citizens before you start having any real problem. If you keep expanding, you'll easily have access to way more than just 2 luxuries, plus the likely possibility of getting amenities from certain city states, namely Muscat, Zanzibar, Cahokia and Buenos Aires.
Unless loyalty is a real issue, which isn't likely, it's always better to expand as much as you can, as fast as you can, without giving much consideration to amenities.