So early on in London there were problems with drivers not having proper insurance to carry passengers, for one, so if anything went wrong there would be no cover (and the driver was technically driving illegally as insurance is required). In some places (again like London) there were also requirements for background checks and registration before someone could be a driver, which uber worked around or just ignored when they entered the market.
Okay. Not having taken London cabs, I cannot disagree with you. What I can tell you is that Ubers in NYC are much safer than cabs in NYC, and Ubers across Africa are much safer than their cab equivalents. To the degree that Uber's technology actually achieves something that cab companies cannot, it's punishing bad driver and rider behavior via the rating system.
These are just two examples from one market.