The Uber network monopoly effect is on the seller side, not the buyer side. It's having a large majority of the drivers under your control that makes it work, because the biggest seller is most likely to have a car nearby.
If I was going to take on Uber I would wait until they IPOed and then build out a business based on local dominance in secondary markets where I would concentrated on getting the transport and maintenance costs down as much as possible. I would lobby the local governments to make life difficult for Uber while promoting the “local hero”. Done well this strategy would be hard for a large player like Uber to fight.
I think in major cities, both Uber and Lyft are becoming commodities - both of them have ample enough supply that I just choose the cheaper option.