I'm curious about this approach, as I always go all out to claim as much territory/resources as I can, as soon as possible. I'm pretty new to the game though, so I'm still learning! Can I just clarify please: are you saying that each of the first 9 citizens needs to yield 2+ production each?
Lets say you settle on a plains hill (2F2P). Also, lets say your first 6 tiles around the city have 2 tiles that are 2F2P (jungle plains hill or forest grassland hill).
That's 6 food 6 production with 2 citizens. A monument in 10, a scout in 5, a warrior in 7 turns. A settler costs 80 production. So with 6 production, that's 80/6 rounded up = 14 turns.
Without spending money on land claims, your auto-culture will grab resources first. These tiles might be what you want (say a bananas jungle hill) but are usually not.
Once you run out of 2+ production tiles in your capital, it's highly inefficient to work tiles that provide 1 or 0 production. What good is growth if you have no use for it yet?
So now it's time to expand. If you time it right (and you should), your capital will grow to size 3 and you'll pop the first settler on the same turn, so it jumps back go pop 2 again.
Now that you have your settler, it's time to move it as quickly and efficiently to the next spot. Again, you're looking for a spot to repeat the process.
If you place it right, your new city will also be on a plains hill be able to work at least one or two 2F2P tiles. This brings the second city's efficiency on comparable levels to capital, as it, too, will produce 4F4P and then 6F6P per turn. Your first expansion (or second city, however you take it), has a 80 production
debt it has to pay off. For the same production, you could've gotten 2 warriors, which equals perhaps two camps cleared (80 gold + 4 to 6 era score), or could've met a city state or three.
If you place your city in a fashion it yields 2 production for a long time (flatland placement, working a flat tile), it would take it 40 turns (80/2) to pay off the investment. But, If it yields 4 production from the turn you place it, it will take 20 turns to pay off this debt and start contributing to your empire.
If you have really good tiles to work around your capital, it might actually be more beneficial to grow to the production cap (say pop 5) and produce things that benefit you immediately (warriors, scouts, monuments whatever it is), especially if you have a really good tile for growth available (say floodplains rice) in the first or second ring.
Now imagine this process for 3 cities total, your capital and two expansions that have 3 population each. You'll be sitting at 18 production per turn without ever needing a worker or monument. Lets say get your first expansion settled at turn 20 and your third at turn 30.
At turn 50, you have three cities that realistically have enough production to get whatever you need, and quickly.
For example, you could get warcarts churning out 3 at a time every 6 turns. That's 6 warcarts in 12 turns, lets say it takes 10 turns to reach your opponent. The first AI is dead at turn 82. And now you're sitting at 6 cities. Probably double from everyone else at that point. If you attacked two AIs simulatenously, you're looking at 10-12 cities by turn 100. Remember, all warcarts you needed were produced in mere 24 turns (3 per 6 turns for a total of 12 warcarts). So by turn 74, your cities are already doing other things, like setting up districts, building workers etc.
Long story short, 3 production per turn
now is worth infinitely more than 3 production per turn 20 turns later.
(this was a problem in civ5 as well, many strategies revolved about worker stealing from CS and stuff like that, but good luck doing those things on low difficulty when they put out their first worker too late)
