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Even on Prince difficulty, for my experience, taking cities with loyalty takes so much time and asks so much.
I see happiness as a support for war. Ecsatic population will let you battle longer and reduce trouble on conquered cities.
And ofc, big population can provide you more resources and production that can help on any type of victory.
Also, keep in mind that ecstatic population will grow faster. And the more it grows, the less happy is will get automitacally. If you want to keep them ecstatic, you need to control the population growth at some point (or recruit Great Merchants)
1 luxury will only serve 4 cities(even if you have 10 copies of that luxury). There are only so many luxuries in the game so you will run out.
You can pick up some additional amenities with entertainment districs, water parks, and certain policy cards but it still wont be enough if you want to have a lot of large cities.
There are a few options you can try though.
You can try to stop housing on some of your cities at 7 or 10 population.
You can try to not build quite as many cities.
You can check the workshop for mods that do things with resources or amenities.
Resourceful mod adds more luxuries to the game so that mod might help you.
One thing to note: We probably have different ideas of what a large number of cities is and what large population is. When i say a lot of large population cities, im talking about 40+ cities with 20+ population.