I wouldn't be surprised to see Uber riders blocking taxi ranks if Uber actually stopped operating, Black Cabs serve very few Londoners.
Edit: Not "literally couldn't get home", I would just have to spend over an hour on a night bus.
Another huge reason is that black cab drivers need to pass The Knowledge, a test of London streets, that often takes 3-4 years of study to pass, and AFAIK the requirements for this test haven't changed in the age of smartphones. This is utterly absurd with modern GPS, and just serves as protectionism and a false barrier to entry for new drivers.
Perhaps it's a regional condition?
I'm not being flippant here: customers have routinely shown that for lower prices they'll suffer almost anything, to the point that safer alternatives become uncompetitive.
TfL isn't asking for much here: just that drivers are vetted properly and consistently. Every other firm in London is asked to do it, and complies. If Uber can put them all out of business by saving money on not doing that, the market will make the industry less safe for passengers.
We have voted in people to ensure that doesn't happen, because we want protection from the free market, simple as.
Do we have any numbers on sexual assaults by Uber drivers in London?
Perhaps it's the London / out of London divide. The BBC certainly has a London-centric approach to news on its all-day news-recycling channel, I noticed.