And what assurances do you have that the Taxi Driver in the Cab you ordered isn't a serial rapist or murderer, or just plain nuts. At the end of the day you take your chances.. Taxi companies have more incentive to "eliminate" people because they have a limited number of slots with a large number of applicants.. Uber (and other ride sharing apps) don't exactly suffer from saturation of resources.. so their checks are nominal (they do background checks, do they ID verification).. So its quite a bit more secure than a "gypsy cab" but slightly less than a formal taxi company.
My issue with the taxi companies is their refusal to adapt and grow.. Rather than adding support (without massive surcharges) for CCs.. they fight it.. (eliminate cash from the equation and a lot of the taxi assaults and robberies go away. Rather than having a more modern booking system, they still use the old switchboard model.. Rather than using the tracking they already have on some taxis and allow it to be expanded to those requesters.. they close door it. (lets not even get into the Medallion fees, and other items)
But statistically speaking, your chances of running into a murderer driving for uber is the same as running into a murderer as a Bus Driver, or in a Bar. And as a passenger, your position is not "I'm in a taxi, I'm safe".. it should be "I'm in a strangers car, I should be mindful of the driver and my surroundings". Now when robotic (non human) taxis come about, then get cozy.. (I welcome our robotic overlords.. because humans are just problematic) but until that time, stop worrying about a few bad apples when these same bad apples can/do exist in other transportation options (except air).. and I should point out.. MOST of the danger is NOT from the Driver.. its the passengers.. Taxi Driver have the highest murder/death rate of any profession.