My fear about overlap isn't irrational. At worst overlap and mountains would mean city has only handful of tiles it can work. City would stay small, its resource value likely low (depending on surrounding resources), and existence of that city would drive up city maintenance cost. As I understand, city maintenance depends on total # of the cities and distance from the palace. In addition city needs military unit to garrison it to keep population feel safe and happy, and that just drives up the maintenance/upkeep cost.
With that in mind, I prefer to have cities which have much room to grow, have many tiles to work on, and can have good production. That keeps city # low, city maintenance low, need for military lower and less settlers to build. Cities with good production values don't need to specialize: they can build everything and excel in almost everything. I have 3 exceptions: cottage farm for creating commerce (Ravenna is for that purpose), securing important PRODUCTION resource (oil or the like in middle of the desert), and integrity of my civ so other civs don't divide mine in the same manner as Poland divided Germany at end of the First World War. So in short: I want my cities be few but large and effective all around which can build up military in short order as needed. I don't see a way to do that with large number of cities in close proximity with significant overlap.
Okay, look around in your empire, and tell me what your average city population is. 4? 5 if we're generous? Rome, your biggest city, has 8 population, and a bunch of them are unhappy. We'll be generous, and assume that you can get every city up to 10 population after Monarchy without having to go out of your way to invest into happiness and health. A BFC has 20 tiles. That means that it quite simply doesn't matter if half of a city's tiles overlap because it can only work half of all available tiles anyway, and even if some more overlap you can employ some specialists, plus Slavery is always an efficient way to deal with surplus population.
But let's talk money. Sure cities cost money, but they also make money. After Currency (a very important technology) you can expect every city to make
AT LEAST 2 Commerce per turn from trade routes from the moment it is founded. Then there's an additional 1 Commerce per turn from the city center, so that makes at least 3 commerce per turn for every city just for existing, likely more because international trade routes give double yield and with all this awesome riverside land you have it's very likely you'll work some more with the city's population, but I'm only talking about the absolute minimum here. Granted, that absolute minimum amount of commerce you are practically guaranteed will likely be canceled out by upkeep until you get a Courthouse up. Also IIRC if the city has at least two population it will automatically grant free upkeep for one unit so its garrison of a single Archer will become free pretty quickly, and after you grow to size 4 it will start decreasing the upkeep of other units. Sure a city is an investment initially, but in Civ4 they start paying back relatively quick even at pretty small sizes.
Let's look at that future deer city north of Antium for example:
There are only two workable tiles it has access to that no other city does, a forest grassland Deer and a regular forest grassland, both on a river. However I sincerely doubt that the neighboring cities Antium, Neapolis, Berlin etc. will grow beyond size 10 anytime soon (never mind that you still have to conquer/settle some of them), so we can safely assume that this Deer city can also work the Iron grassland hill and two riverside grassland tiles without really infringing on another city. At size 5 with basic infrastructure (Granary and Courthouse) in place this city would yield 7 production and
at least 9 commerce per turn assuming cottages on all three river grassland tiles, but at that point they are likely all hamlets already so we can safely be talking about 12 commerce per turn. Even if that city would never ever grow beyond that size 5 (and the three tiles it "steals" from other cities are really negligible because it would take forever for those other cities to grow enough population to work them anyway) and we disregard all buildings or civics modifiers, 12 commerce per turn (which will grow beyond 20 as the hamlets grow into villages and towns) easily cancels out any increased upkeep cost that city might incur directly or indirectly, and even without whipping, 7 production per turn in pre-industrial times is nothing to scoff at. That's a Praetorian every 6 turns.
Now the situation would be different if the 5 tiles the city would work were all naked plains without river, forest or resources, but the map has blessed you with awesome land, grassland, rivers, forests and resources everywhere, and it would be silly not to exploit that to your advantage.
Now that we've gotten cost out of the way, let's look into health and happiness!
Let's be generous and assume that because of resources, Hereditary Rule and terrain like rivers and forests any city you found won't encounter happiness or health problems until it grows past size 10, without you having to actively work to increasing health or happiness. If you have 20 good tiles, and you settle one city to work them all, you need to invest quite a bit in health and happiness infrastructure to get to 20 population. 100 Production for an Aqueduct, 150 for a Grocer, another 150 for a Market, 80 for a Temple, that's 480 production and even with all relevant resources you still start running out of health past size 16. Imagine that instead you just invest 100 production into another settler, so instead of one mega city with 20 population you only need two average cities with 10 population each to work all tiles, at no extra cost for health/happiness infrastructure.
Also it's way faster to get two cities to 10 pop each than one city to 20 pop because you need more and more food to grow to every additional size.
That room to grow you like won't do you much good if it takes 200 turns to actually grow to take up that room, while with two smaller cities you would only need 50 turns to work it all.
Also you really need to read Adam Smith yo, specialization is the foundation of all human civilization. Instead of trying to build one city that someday will be able to yield 50 production and 50 commerce per turn, I guarantee that you would sooner be able to build two cities, one which yields 40 production and 10 commerce and another that yields 10 production and 40 commerce, and it would be cheaper production wise because the 100 production for the extra settler is nothing compared to the 500+ production you save on health and happiness infrastructure.
Not every city needs good production, especially when it has access to lots of food anyway.
Also 10 size 8 cities can more quickly assemble an army than 5 size 16 cities thanks to Slavery and Drafting.
In short:
"Quantity has a quality of its own."
-Stalin