>However, the more people that use Uber will lead to better service levels and lower prices.
That would be very bad for Uber. Lower prices would mean that they are constantly operating at a loss against traditional taxis and other competitors or apps.
Your argument fails to support that there is a network effect that would raise the cost of switching to alternative means of transport and incorrectly assumes that uber is able to operate at a loss constantly as the network size grows.
Take a look at the airline industry. The first guys to do it are nowhere. Everyone is undercutting each other because customers are not locked into one airline and are price sensitive.