comparing https://www.lyft.com/cities to https://www.uber.com/cities/ it seems like there is a huge discrepancy between the cities that have lyft and those that have uber. Uber has a significant international availability.
1) I'd imagine there are significant costs to entering a new city. Uber is a two-sided marketplace so in any new city Uber needs to invest resources and time into acquiring supply/drivers and to a lesser extent users. Combine this with the operational support to maintain that marketplace and the overhead becomes non-negligible.
2) First mover advantage is a real thing so any ride-sharing network that enters a city with an established player is going to have a harder time acquiring the aforementioned supply/demand.
3) When more than one ride-sharing company is in a city it eventually turns into a subsidy war for demand. From what I have read this is what happened to Uber competing with Didi in China and resulted in massive losses for both companies.
TL;DR entering new markets is in fact pretty expensive, and far more so when there is existing competition.
They also need local support infrastructure. Lyft actually does an in-person interview with every driver, which requires a human being there. They have at least re-used resources by promoting certain drivers to a higher level that enables them to run these interviews. But bootstrapping that infrastructure requires some effort.
Between the costs, the marketing efforts, and the support infra, it's not as easy as just flipping a switch.