Having difficulty getting a taxi is a frequent problem in many US cities. The root cause of the problem is the artificial scarcity of taxis caused by the medallion system. The number of licenses available does not meet the demand. The problem is that once that system is in place and the market value of medallions has become substantial, the medallion holders have a huge vested interest in the system and it is therefore difficult to change. (The market price of medallions in New York City got close to a MILLION dollars before Uber came along and started to reduce their value. Their value has fallen by over 50% and continues to fall.)
I believe there is some need for government oversight of taxis and competing services like Uber, but that the medallion system is fundamentally broken. It should be replaced by a system that issues an unlimited supply of licenses for a modest annual fee - enough to cover inspection of vehicles, background checks of drivers, tests of taxi meters in vehicles that continue to use them, assuring that all vehicles are covered by adequate insurance, and resources to investigate and act if there are problems with issues like disabled accessibility or racial discrimination. Such licenses should be issued regionally rather than by individual cities and towns. Drivers of existing taxis as well as drivers for new companies like Uber and Lyft should be required to get them.
Uber has fundamentally improved the experience of getting a ride in a few ways. Most importantly, they have removed the handling of money from the in-car environment, removing a substantial risk for both drivers and passengers. Also, knowing the cost of your ride before you enter the vehicle is a plus. Uber pricing is variable because of surge pricing, but you are given a current price when you book and that's what you pay. The cost of a regular taxi ride is unpredictable, because you don't know how long the ride will take (rates are usually based on both time and mileage) or which route the driver will choose to take.