One of the features of Civ 6's design is that as long as you're playing single player, you can play pretty much however you like and it will generally work. And no one way is "right" or "wrong" since it's just about having fun.
Culture is very useful, I agree, but it won't help much if your neighbour gets to Knights before you do and crushes you. Which the AI probably won't do, so you're right, you don't need to stay ahead of the AI in science. It's likely just memories of earlier Civ games that makes me wary about not keeping at least tech parity with my neighbours, which you definitely need Campuses to do at higher difficulty levels.
When you found a new city and can build one district, you can spend that district on a Theatre for culture, and I agree, culture is very useful for running you up the civics tree. Or alternatively, you can spend that district on a Campus to gain science and build a Monument to get culture and you've boosted both.
Once you get to four citizens, you have another district decision to make. At this point, if you're falling behind in culture, i.e. your tech advancement is ahead of your civics advancement, a Theatre may make sense. If you haven't built an Encampment yet, your first Encampment will likely generate even more culture for you over the course of the game from the inspirations it provides. Alternatively, a Commercial Hub or Harbour could be a good choice to get a combination of culture, science, and gold from a new trade route. I like to run the policy card that gives you +1 culture / +1 science from international trade routes, as it really boosts both. If you have good relationships with not-yet-conquered cultural city states, that boosts the value of Theatres to be sure. And if you're playing peacefully and want to ensure lots of golden ages, archeologists are a sure fire way to generate tons of era score, which again is a reason to like Theatres.