This, though, kind of negates your arguments in favor of the Pyramids. An extra policy card that boosts production in a dozen or more cities over 100 or 200 turns could add up to a benefit of 10,000 cogs or more. On the other hand, by late game I've often got cities that have run out of just about every useful thing to make, so I set them to making builders, which they each can do in 1, 2, or maybe 3 turns. At this stage, then, the Pyramids would not really confer any kind of benefit.
I don't understand how you can argue that the Pyramids do not provide any benefit if your cities are able to produce builders quickly. In fact, the benefit is measurable in terms of production saved as I demonstrated.
In addition, I doubt you can benefit from Forbidden City for 200 turns in a typical game. A typical game, in my mind, ends by Turn 250 (Standard speed), and Printing isn't exactly easy to beeline. In fact, the only two reliable ways of gaining the Eureka for it is to build two universities (requires Education) and to obtain Great Engineer Bi Sheng (requires building industrial zones, and ergo unlocking Apprenticeship). It's hard enough to unlock Apprenticeship by Turn 50 (requires 444 science, including Mining, after hitting all relevent Eurekas). It costs an additional 675 science (including Masonry, after Eurekas) to unlock Printing. In that time, you'll also need to build at least one industrial zone and generate enough great engineer points to recruit Bi Sheng. Even if you could somehow unlock Printing by Turn 60, you still need to put 920 production toward the wonder itself. It would be an absolute feat if you could build Forbidden City by Turn 70. If you're at this pace, you probably won't need more than another 100 turns to win the game.
In a more reasonable scenario (one where a game takes 250 turns to complete), I think you'd probably clear Apprenticeship by Turn 80, Printing by Turn 100 and still require another 10 turns to build the wonder, so you'd be benefiting from an extra policy card for 140 turns. In addition, despite having to beeline Printing, if you were still somehow able to keep up with your culture yield and hit all relevant inspirations, by this point of the game, you might
have cleared either Guilds or Medieval Faires and have adopted a second-tier government. Considering the 13x ROI of Pyramids I mentioned earlier, Forbidden City would need to bring in what's equal to roughly 10400 production over 140 turns (~74 production per turn). I'm assuming the 5 culture per turn it provides is equal to 120 production. I'm discounting it a bit because the you could conceivably build the Pyramids 50 turns before you can get Forbidden City. Obviously, there are some caveats like how much earlier the Pyramids starts to benefit the player (plausibly by Turn 60) and things like Monumentality discount, but we'll just ignore those here.
What policy card can you slot in at Turn 110 to gain even something like 50 production per turn? Keep in mind this is your marginal policy, meaning it'd be the best of the policies you'd choose not to implement without Forbidden City. I know this isn't an easy question to answer because policy benefits aren't always easy to convert to production, but let's try anyway. I imagine Republican Legacy would be a decent marginal policy candidate. On average, I wouldn't expect this card to produce more than 5% boost to yields, considering a difference of at least 2 amenities is required for a jump in happiness level. Let's say we have 10 cities by Turn 110, and each city yields 30 production per turn and 20 gold per turn. I feel 30 PPT per city is generous to assume at this stage of the game, but I'll allow it because we had to rush Apprenticeship and probably build a few industrial zones to then rush Printing and the wonder construction. 20 GPT * 10 cities - 50 GPT in maintenance should yield 150 GPT, which seems more than reasonable to me at this stage of the game, especially without prioritizing commercial hubs. The 5% bonus leads to an increase of 10 GPT to 160 GPT (= 200 * 1.05 - 50). We'll choose 3 gold to 1 production as the exchange rate. This should be on the favourable end, since usually, it costs 4x production to purchase something in gold. Between gold and production, we're getting roughly 18.5 equivalent production per turn (5% of 300 production + 3.5 converted from GPT). We're going to assume we're generating a negligible amount of faith because we shouldn't even have unlocked Astrology by this point. That means Republican Legacy must yield what's equivalent to 31.5 PPT in growth, science and culture. I understand growth is very important in this game, but we'll ignore it here because calculations become very complex when growth is involved. We just want to see if we're in the right ball park. Let's also assume we're generating 80 science and 80 culture per turn. Do these yields combine to at least equal 630 PPT (= 31.5 / 0.05)? I really doubt it. That's especially so when we introduce the concept of time value (e.g. $100 next year is worth $90 today if you can make $10 on the $90 in a year). I won't get into the details here, but because of the aggregation requirement of production (e.g. you don't get 1/6 of a monument this turn if your city yields 10 PPT; you have to wait 6 turns to get the whole thing), a set percentage increase in production is usually worth more than what you'd think.