Culture's always critical, regardless of what victory type you're going for. You need to get to Political Philosophy quickly, to Theocracy quickly if playing for a religious victory, and getting to a Tier 3 government is often a bottle neck for me if I haven't kept up my culture output while going for a science victory.
@acluewithout has argued that a cultural victory best integrates the various game mechanics. And I supposed that's true, in the sense that it's a relative victory condition, where your success is based on your progress against the progress of other civs.
The whole tourism mechanic seems very silly to me, though, as currently implemented. Lots of tourists come to my country and that has no impact on the game whatsoever, until one opaque counter exceeds another opaque counter and I'm declared king of the world. Sure, Seaside Resorts generate gold, but National Parks don't and neither do Great Works or "tourism on flight" improvements. It's good that Seaside Resorts do, because it gives you more than one reason to build them, and I dislike improvements that exist only for pursuing a single victory objective/yield. So yes, Seaside Resorts are good, but the whole tourism mechanic, not so much.
I agree, too, that the whole raising Appeal can be a fun part of managing your late game empire. It creates an opportunity cost to Mines and other things, and generally opens up a new avenue for game play. But again, it's so loosely tied to other mechanics, that it's really only relevant when going for a cultural victory. Why isn't it tied to Loyalty, or Health/Housing, or even Amenities (high appeal tile = 1 Amenity, for example)?
In Civ 4 and Civ 5, cultural influence was much more than attracting a bunch of tourists. It was winning over the hearts and minds of other civs. It wasn't implemented perfectly - not even well, really - but it had some impact, potentially forcing a government change in Civ 5 or flipping a city in Civ 4. Loyalty right now is a less interesting city flipping device, being based primarily on relative population, but could be a better one if cultural influence was integrated with it for a combined effect.
Hopefully we'll see some improvements to this system going forward. If not, then you can always just capture the AI's cities, at which point your empire magically converts to a single culture since there are no multi-ethnic societies in Civ, and win a cultural victory over the last remaining AI city.