A bigger question here is should part-time, low-paid freelancers like uber drivers be allowed to force out full-time taxi drivers by eating into their business?
It's a small, but I tend to think it important, difference but you really should look at the premise as "whether we should restrict people some from being allowed to offer the service in order to protect medallion owners and taxi drivers?". You don't need to do anything to allow something, so the question should by default be whether we should be doing anything.
As to protecting taxi drivers. Firstly I'm not entirely sure that it is Taxi drivers suffering the most, that would be medallion owners, who have shown they only care about protecting their profits not customers. Secondly, progress requires that some roles become less attractive or even cease to exist. Lamp lighters, stable hands, farm workers etc were all massive sources of employment prior to technology making them largely obsolete. Taxi driver is just one of many jobs slowly following the same path. Personally I think autonomous vehicles will almost entirely kill the field within a decade anyway, so Uber is the least of their issues.