Well, "Quick" speed is kind of a misnomer. After all this time, I've only finished a single game. Even at the fastest speed, on the easiest difficulty, it takes ten hours to go from start to finish. I suppose I could have gone for a quicker conquest victory, but I wanted to see the whole tech tree (also, I don't like doing that generally).
I enjoyed myself, but most of my enjoyment came from comparing numerous picayune details to the other games of the series. What do the different World Wonders do, and which ones got added and/or dropped since the last game? How do the different units compare to each other? What's their combat strength and which ones have interesting special abilities? Which leaders made the roster this time (and I think, after six tries, the Civilization series finally got it right by choosing Teddy Roosevelt as the leader of America - Lincoln and Washington may have been more vital to the formation of our country, but TR is the president who was most like a video game protagonist)?
And if I'm being perfectly honest, a lot of the stuff I think of as "big changes" are probably also tediously small details to anyone who's not already an obsessive fan of the genre. Like, you have to build important stuff in the tiles surrounding your city and not just in that nebulous "city interior" now. This makes a dramatic difference in the high-level strategy as you juggle adjacency bonuses, space limitations, local resources, and your society's needs in order to get the most out of your limited number of turns. However, it is also still just picking stuff out of a list in an order that will make various numbers go up most quickly. If you weren't a fan of the game before, nothing about the new system will make you change your mind.
Likewise with the change to city happiness (it makes building a specialized entertainment city profitable now) or social policies (they're now laid out like a second tech tree, which has some interesting philosophical implications that I may have to talk about in a second post because they're ideologically very complicated) or city-state relations (you can't bribe them with gold any more, which makes keeping them on your side a bigger challenge). If you're committed to the idea that "shuffling numbers around in various menus" is a big enough playground to house multiple, very different games, then the changes are enough to dramatically alter the way you approach the game. On the other hand, it's still just a Civilization game.
I can't even really say it's better than its predecessors. The base game feels more confident in its mechanics than Civilization IV or Civilization V did, pre-expansion, but there's no longer any circumstances where you'd play Civ4 without Beyond the Sword or Civ5 without Brave New World and Gods and Kings, so there's no call to judge the prior games as anything but the best versions of themselves.
The transition from Civilization IV to Civilization V was something of a lateral move. Both games had their flaws, and aside from super-long turns in the late game (something no 4X has ever solved), there's not much overlap. Neither game is a replacement for playing the other. And for now, that's true about Civilization VI as well. The geometrical puzzle of laying out your cities is fun in its own right, but no replacement for the compact, streamlined board game experience of Civ5 or the super-detailed, almost sim-like mechanics of Civ4. Who knows, maybe with a couple of expansions Civ6 will be the benchmark against which all other entries in the series will be judged, but for now, it's just another step sideways.
(Which, you know, is fine. I love the series, so "as good, but not noticeably better than what came before" is a very high recommendation in my book).