The cities certainly WERE high pop cities in the livestream... 20-30 pops in antiquity... So maybe you're onto something. Of course, we'll have to wait quite a bit to confirm
There's a possibility that the game is currently wildly unbalanced, and population isn't supposed to grow that fast, but assuming that's not the case, I think this is how things could work:
In Antiquity, cities are able to grow very quickly from abundance of food from rural districts, aided by food supplied by towns. The capital, for instance, should be able to reach 30+ pops by mid-late game, and this will be enough to essentially saturate all of the workable tiles in the city. This means that farming towns, while useful early-game, will lose their significance quickly and naturally become first candidates for city promotion. Mining towns, which provide extra gold, can be used to accelerate the promotion of farming towns by helping you collect the gold required for promotion more quickly. In the meantime, farming towns can be put back on growth focus, so that you can save on promotion expense. Once farming towns promote, value of gold diminishes since gold's primary use is to develop and promote towns. That puts mining towns next on the list for promotion. Now, as more towns promote to cities, there are fewer towns left to support the growth of new cities as well as the push for promotion of older towns. This is when it would make sense to settle a new town, and you'll want to try to time your next settlement limit increase (obtained via civics, etc.) to coincide with this.
In Exploration, I'm assuming the number of specialists each urban district can host will increase (probably to 2). I don't know if that increase happens with the beginning of the age, with a certain tech or civic, or with new buildings available in the age (this is the closest one to how specialist slots work in Civ 6). Whatever it is, when it happens, it will do two things: a) cities are no longer saturated at ~36 pop, and b) urban districts become radically more valuable. The latter factor incentivizes you to start irreversibly replacing rural districts with urban districts. That means your capital, for example, will actually start struggling to support its own growth. In terms of metagame, all this could imply one of the following:
- You should go back to settling more farming towns in Exploration.
- You shouldn't have built any farming towns in Antiquity, as they're inferior to other towns. Instead, you've started building them in Exploration.
- You should've kept your farming towns from Antiquity instead of promoting them. When they're done pumping food into cities, put them back on growth focus for the rest of Antiquity, so that they're bigger and can thus export more food in Exploration.
Regarding the other two town types:
- Fort towns are very special towns you'll want to use for supporting your war effort and never for supporting your cities. I imagine that will be the case in all three ages.
- Trading towns, since they provide happiness, probably grow in importance as time passes and your cities fill up with specialists each of whom incurs a -2 happiness penalty. It also thematically fits for trading towns to start becoming useful in Exploration.