TL;DR: Taller civilizations are now much harder to pull off in Civ VI. Instead you are encouraged to go wider.
You question stretches across a large number of the changes between V and VI, so let me take them point by point -
Global happiness was removed
This is true, but the concept is not entirely gone. Happiness was replaced by the Amenities system. Each city now requires a certain amount of amenities based off its population, but if one city doesn't have enough amenities it doesn't affect the rest. Amenities come from 4 general sources:
- The Entertainment district, buildings, and wonders
- Religion
- Government policies
- Luxury resources
That last one is the most important for your question. Each luxury resource provides 4 amenities (meaning having duplicate luxuries does actually have value), but each city can only use one of each type of Luxury. The game is balanced such that you can't really have your cities survive off the first 3, you need luxuries. Since luxuries only give 4 amenities per resource though, that limits your growth. Also don't forget that just like in previous civs, unhappy cities (aka not enough amenities) can spawn rebels.
In addition, science and culture costs increased for each new city you
founded.
I haven't confirmed this, but I'm pretty sure this is no longer the case, or at the very least is much less extreme. I can tell you that the culture victory no longer lends itself towards smaller, taller empires. Instead it too encourages a large thriving empire (its a similar tourism system to Brave New World).
What mechanisms exist in Civ 6 to penalize you for creating a wide empire?
Luxury resources is the big one. On top of that some civs now explicitly don't like it if you get too land-grabby. You also have to build a settler for each city, and settlers now cost 1 population each (instead of halting all growth in V). You also have to be careful in managing your finances when you have a lot of cities. Remember that most buildings have maintenance costs, so if you go full science with no finance you will go bankrupt fast.
I think though there is one question though you are forgetting to ask:
What mechanics exist to penalize taller empires and encourage wider empires?
The big one here is the city building system, specifically Districts and Wonders. They each take 1 tile now, replacing the effects of the old tile. That means you can no longer have one uber capital that pumps out 32 wonders in 3 turns each throughout the game. There just isn't enough land for it.
Following on that, Wonders (and Districts) also now have terrain requirements. If you want to build Petra you actually need to have a desert tile in your city borders.
Finally, all the victory methods now discourage small empires. Obvious domination and science always encouraged large empires. Religion also encourages large empires because you need a lot of faith to pump out religious units. Faith only comes from religious buildings, so you want a lot of them, meaning you need a lot of cities. The big one though is cultural victory. Just like Brave New World, its all about tourism here, meaning that you need lots of great works, relics, artifacts, etc. You need to store these in museums and such, and that means you need cities. The more you have the more tourism you can generate.
Last point - its important not to forget that having a lot of cities is just a lot of work... Turns start to take forever and the game becomes less fun in my opinion. Its not really a game mechanic, its more of a flaw or even an opinion of mine, but its something work considering.
With all that being said, you are still encouraged to create good, productive cities. One of those is still worth multiple small cities
Best Answer
Korea is exclusive to the Rise & Fall expansion pack for Civilization 6. You probably do not own this DLC.