Yes, you're very right about The Wheel... I hadn't noticed that.
The capital has 0 connection to rivers as it is. My bad

So The Wheel is needed, probably right after Agriculture, as has been advised. And then Pottery is a very cheap tech to get.
I still don't think there is immediate use for cottages but the cost for Pottery is minimal.
The mines are 4 yield tiles + commerce vs 3 yield tiles + commerce for floodplains... for the purpose of building workers.
Working 2 mines + corn + oasis = 12hpt towards worker
Working 2 cottages + corn + oasis = 10 hpt towards worker
Now... being IMP...
Working 2 mines + corn + oasis = 16hpt towards settler
Working 2 cottages + corn + marble = 12hpt towards settler
You're right that just getting the maximum amount of hammers isn't necessarily what matters.
Ensuring the commerce output is good as well.
However, with a trade connection, the first two cities can
raise the commerce output instantly (provided they're not too far and/or can work some riverside tile).
This is without requiring even an improved tile.
So, getting the first (two) settler(s) out asap is actually pretty big.
In this light, improving mines instead of cottages makes a lot of sense (to me).
ps : which is more :
Settling earlier cities means they can produce earlier workers, which could end up with a faster cottaging - or at least allow to catch up with a cottage first approach.