You're not arguing for deregulation in urban areas, you're arguing for less-stupid taxi regulation. I agree - taxis were and are awful. They were an industry ripe for disruption. But that disruption should be along the lines of "this medallion allows you to dwell in this zone for this many hours a day not including 4-6pm and therefore costs 10% what a full one costs."
I'm arguing for safety realized through convenience. That could be satisfied by taxis or by competing services. The thing I know is that there is no way before a couple years ago that I could get a car on demand within 10 minutes at 1:00 am on a Tuesday. Uber's regulatory conundrum notwithstanding, I would like to continue to be able to get that car when I need it.
Taxis are regulated. "competing services" are not. I will happily agree that the companies that operated within that regulatory framework were ossified and anticompetitive but that's an operational problem, not a structural one. Your 10 minute 1am car is earning its driver $3.50 an hour and costing its company five billion a year. Yes, it's convenient but it also isn't steady - state. The argument here is that venture capitalists are underwriting a Dickensian lifestyle for the drivers and fucking up traffic for everyone else and should Uber's note holders decide they're sick of the experiment, we'll all be paying market rates once more.