You get what you pay for. If you're happy to entrust your life into a random untrained uninsured stranger, then go ahead and get the cheaper unregulated taxi.
Regulations didn't save John Nash from dying in a traditional taxi. If you're happy to entrust your life to a random stranger that's shielded from market conditions/feedback, then go ahead and get the regulated taxi.
You think market conditions will make taxis safer? Why do you think we have OSHA? Why do you think selling products with lead paint in them is illegal?
Market conditions have made transportation safer. Two way rating systems, accessible GPS, drivers #, drivers picture, driers location, arrival time etc.
These qualities are present because they were superior to the previous market provider, in order to compete in a free market your product or service has to be superior in some manner. Competitive market forces like safety, efficieny (both price and liquidity) are what makes Uber successful.
No laws dictated these qualities, no legislation demanded this level of safety. These practices were conceived via market forces, and safety was a by product.
The entire point of Uber is that the taxi industry is not a free market, therefore the things necessary to compete in a theoretical "free market" are not relevant and can't be assumed to be why Uber is successful.
Uber uses roads that were designed to comply with government safety regulations and their drivers drive cars designed to comply with a different set of government safety regulations. Those regulations exist because experience has shown that they need to.
Safety precautions are in fact codified by law. Uber is in the business of breaking the law and ignoring all regulations. The fact that they have a substitute that follows the spirit of the regulations if not the letter and provides safety is the only reason governments have chosen to allow Uber to exist. The safety precautions Uber has ARE required by regulation, though implicitly; they act as a substitute which makes Uber's lawbreaking far less egregious.
I'm pretty sure that the taxi that was being touted for the first time was also unregulated, so it's extremely unclear here what 3x my money was going to get me.
Quality of black cabs and mini-cabs in London in also wildly variable. At least with uber you can actually leave a bad review (I'm not sure what comes of those) but I get a response on bad reviews at least.
Almost every black cab ride I've ever had has been great - and they most certainly know how to get round London better than a newly minted uber driver with google maps. However if you've never had a black cab with the driver spouting racist/misogynist/anti immigration rhetoric they whole journey, then you haven't lived ;-)
A lot of taxi drivers grew up when London was part of England, before it became a "global city". It's unsurprising that they miss those days when it was still culturally part of England.