I know everyone is all over Uber and and the other one because the cars are "nicer" and the service "better" than cabs. But [...]
Um... isn't that enough?
Firstly, you're wrong about the liability.
Secondly, you are confusing the possibility of injury with its probability.
If the probability of injury is small and the cost of injury is also appreciably small, the expected cost of using Lyft or Uber may be much less than the expected cost of using a cab.
For an example, if a ride-share is $6 less than a cab fare, and if there is an average of 1 injury every 100,000 rides, then if the average injury costs less than $600,000 then it's a better deal for everyone to use the ride share.
Using this reference, cabs crash about once every 300,000 miles.
Also note, the number of crashes in regular driving has decreased dramatically over the last few years, probably due to increased safety measures in vehicles and modern roadway improvements (Denver Barriers around bridge supports, for example).
And in any event, most people have health insurance. At the very least, a significant portion of riders would have health insurance - enough to reduce the risk by a further factor of four or more.
SHELL GAME is where you can't win. CASINO GAME is where the odds are against you. Uber and Lyft seem to be decidedly in the passenger's favor.
Cue the irrational fearmongering reply: "unless you are the one injured, then how would you feel!".