Assuming they don't, perhaps it is not as effective relative to the cost as you might surmise. Perhaps for that reason we ought to be skeptical of any government, such as the Austin City Council, presuming to know how safe, comfortable convenient transportation is best organized.
And this assumes that the council wants what is best for potential drivers and passengers as a whole, not what is best for the existing taxi cartel. Needless to say, that assumption is very questionable.
I’m just thinking about it from the perspective if I had to insure Uber against misbehavior of any of its drivers. I’d want a basic background check at a minimum, and that would include fingerprints.
I am assuming that Uber has insurance. Maybe they don’t.