Then consider what a huge nation in a developed game can do to get more culture by ACTUALLY using the slider for something other than research!!! *gasp* Is it worth it? When you're talking about the power you get from a Tier II or III trait, it may well be. Playing FOR trait selections is intended to be a strategy that can compete with research, and it may take knowing when to let off the pressure and when to best put on the pressure and having the strategic cleverness to withhold some effort in that direction until later in the game might enable you to play to go HIGH with traits as opposed to WIDE with traits.
First, I fully approve of wanting to make it worth using the sliders for things other than research. And indeed, I might have considered doing that - the traits do a good job of encouraging that, at least a bit. (In my own game, I haven't used the culture slider. Not that I've actually unlocked it yet, but even if I had I wouldn't have used it. But I did build Captured Fire specifically for the culture bonus, and have prioritised culture a bit.) However, the higher tiers being tied to tech level gives an incentive to avoid culture output, which undoes that effect rather badly.
You say nothing in the game is like this but city spreading strategies already contain this same duality of interesting choices (tall vs wide) as well. No, nothing QUITE works like this, which is part of what gives it its own character. You may be the first to note some of the intricacies of strategic consideration that I intended players to have to find for themselves.
"Tall vs Wide" isn't the mechanic I was talking about. I'm saying that nothing else in the game actively penalises you for generating resources. Making new cities is expensive (both in terms of building the settler, and in terms of the hit to your economy while the city is in its early stages). But once you've paid that cost and the city is being productive, you won't suffer any further harm from having the city. At absolute worst, you might end up deciding to abandon the city to e.g. settle one elsewhere once your technology has advanced and you've revealed new resources or something. You've wasted the resources you invested in the city, but nothing further has happened. On the other hand, investing in early culture permanently blocks you off from getting as many higher tier traits.
It's hard at the moment to guess exactly where we'll be in the late game with trait accumulation until someone gets there and experiences it. It seems like it gets prohibitive but with so many wonders doubling the base culture output after so many years in play, more techs opening up more powerful cultural buildings, and all of us being so used to culture being something to downplay as not highly valuable after our cities get to the final rungs - this introducing a very different way to think of it - you might shock yourself at how much culture you can plug out if you're still caring about it later into the game. I don't intend for all of your traits to go past Tier I, just a few at most throughout the whole game. The rest would be a matter of finding great synergies between lower end traits to build your platform of mods upon.
Well, this is something where looking at things mathematically will probably make things clearer. (And I'm a mathematician IRL, so that's kind of my default approach anyway.) I'm going to try to analyse what will happen during the game depending on when you do and don't focus on culture, and to simplify the mechanics down. We'll end up with a more abstract mechanic which, on the surface, looks totally different, but is actually just the same mechanic simplified and expressed in a way which is easier to understand.
Now, even if every wonder had its culture output doubled immediately, once the map is filled out the culture you gain per round is going to increase pretty linearly - or possibly quadratically, if there are enough +

On the other hand, if you suddenly start focussing on culture at some point midway through the game, then (again because of the scaling nature of the costs) you'll find that you rapidly make up most of the lost ground from not focussing culture early. In return for making the investment of building lots of culture, you'll quickly end up about a trait ahead of everyone who isn't focussing on culture, and only negligibly behind people who've focussed on culture from the start of the game. (The culture producing buildings you didn't make before are now either obsolete, or very cheap to build. You'll be missing out on the wonders which are designed for culture, but there aren't that many wonders designed to boost culture which don't go obsolete, are there?) And if you later stop investing in new cultural buildings, the people who didn't invest in culture in the first place will quickly catch up with you and you'll be equal with them.
This means that at any stage of the game, you'll have about n traits if you've been focussing on culture recently, or n-1 if you haven't. And at some stage - at an educated guess, quite a while before the end of the game - you'll effectively stop getting any more traits.
So the simplified, abstract way we can look at the mechanics is this: You get a steady sequence of traits over the course of the game, up to a certain point at which point the trickle dries up. At any time, you can get early access to the next trait in the sequence by focussing your civilisation on culture, which you then lose access to if you stop focussing on culture. Of course, this is a simplification: it treats "focus on culture" as a binary choice, assumes it's always worth exactly one extra trait to do so, and ignores the delay between changing how much you're focussing on culture and getting/"losing" the trait. But it shows the key principles, and is a good deal easier to work with and see if there are going to be problems with different ways of unlocking new trait options. The rate of gaining traits will also go down as the game progresses, but that's irrelevant to this model as it turns out.
If the traits are all available from the start of the game, the mechanic is fine. We just need to make sure that the cost of gearing your empire towards culture for a turn is roughly equal to the benefit gained from a turn of having an extra trait, and all will be well. If the later traits in the sequence become more powerful - or offer better options for some people's strategies - then again, this is fine. Indeed, it could even be a way to account for the fact that culture is far more important for other reasons when you're just starting and don't actually have much of it.
But if the traits available improve over time based on some outside mechanic - technology, the number of turns played, or whatever - then we have a problem. At certain points in the game, in order to get access to the next trait, you have to lock it in as one of the weaker ones when it could otherwise have been one of the new ones. Which means that for most of the game, the cost of getting access to an "extra" trait is that you have to focus on culture. But at certain points, the cost is that you have to focus on culture and you end up with a suboptimal trait line up in the future. Since these two costs are very different, it's going to be impossible to balance the traits against them both - either they'll be too strong to give them in exchange for just focussing on culture, or they'll be too weak to be worth getting in exchange for the two costs put together. (In my opinion, at present we're in the latter situation, which is probably the better one if we have to pick one or the other because it's generally better to be too stingy than too generous with new options. But it's still by no means ideal.)
Things are made worse by the fact that the mechanics aren't presented to the player as obviously as I did it there. Because of the inertia I mentioned above, and because it's not obvious exactly when the traits will come if you don't focus on culture or if you do, it's hard to tell whether focussing on culture will pull the next trait forward so it's ahead of the tech unlocking new options. You might say this is an added layer of strategy, but it isn't really. There's a simple correct answer, and you can calculate it by checking the time it'll take you to reach Ren./In. Lifestyle, checking how much culture you're making per turn and how much more you need, and then figuring out how much your research and culture will be increasing over the next turns. That's not a strategic choice, that's a tedious calculation. And one which you have to make quite a way in advance, since if you get it wrong you won't be able to stop the trait being given to you early.
I think players get used to the quick few they get right up front and are hoping it won't get too tough to get the more advanced ones later but they are meant to be very difficult to get. It's designed to be somewhat like military ranks. You might get to e5 (5th rank) within 4 years of dedicated service, but you can very easily only get to be as far as e7 (of 10) by the time you retire at 20 yrs.
Well, I can't say I know much about the US military's ranks, but that also reminds me of the XP mechanics in my D&D campaign, where the cost to gain a level doubles every level like in the oldest versions of the game, and which was designed to do something similar. Then my players ended up spending the first arc of the campaign dealing with a spider demon that massively outclassed them rather than looking for treasure and thus got stuck at level 2 for ages, but I digress. But neither in my D&D game, nor (I assume) in the US army does the amount you gain from a promotion depend on how long you've been playing/in the army.