Speaking of waste, although not entirely related to having a river...
If you build strictly on a grid you're "wasting" at least 8 tiles per city until you develop hospitals, and then you're wasting worker time cleaning up polution until you develop mass transit even if you have no factories, which also means that poluted tile is wasted until you clean it up.
A grid over mountains, hills and desert (without floodplains) means you're wasting tiles that will never be worked by cities that won't grow to at least size 20.
Any city over size 20 is wasting productive population since they'll never have more than 20 tiles to work and at those sizes you'll actually need entertainers, more waste.
4 small cities occupying the same space as 2 large cities will essentially have equal production. Although it will take longer to build individual items, you'll be building 4 at a time rather than just 2. This is a big advantage in the early game and those 200 spearmen can eventually be upgraded to 200 Mech Inf. So it's really 6 of one or a half dozen of the other. The exception is wonders, but that problem is solved by having a few massive production cities that do have their full 21 tiles to play with.
The further away from your empire center you get the closer you can pack them. At "total" corruption levels, a city with 20 pop is no more productive than a city with 1 pop.
Small cites are easier to manage and easier to keep happy. If no city has more population than you have luxury resources to keep them happy with, you don't need to waste comerce on luxury.
And, last but not least, you can always abandon "extra" cities once you have the tech to let your cities grow. So, you can have your grid with extra cities packed in between the grid cities, when you get hospitals you can start abandoning them
Just my $0.02