But you get charged only $10. Uber has already eaten $5 of the fare for you. That money has to come from somewhere. The 30% they take on the $10 isn't covering what they've already knocked off the price.
That money has to come from somewhere. That somewhere is the VC.
The 30% they take is really only stemming the bleeding.
Your mistake (I think) is that you are connecting the costs to the prices on a per-trip basis.
In the hyper-growth phase where the objective is to gain mindshare and market share, their VC and IPO cash can be used instead of charging reasonable prices.
For your scenario it’s entirely possible that the price of the ride is $10 but the cost of the ride to the driver is $12.
Hubert Horan is an analyst who goes into these costs into detail. http://horanaviation.com/Uber.html
I'm not the one telling you that the $3 doesn't cover overhead. Uber is. Taxis that cost more are.