Most taxi laws here require a taxi to be a dedicated taxi and not a personal car...
That's maybe why Brits don't see the problem, here we have two very different things; taxis and "private hire". I've used taxis very rarely and then only when I can bill my employer for the journey - they're very expensive. The good thing about a taxi is that you can flag one down on the street. Private-hire cars have to be pre-booked (although most will let you phone their office to book when you flag them down with a kebab in your hand) and are generally much cheaper.
I do wonder however how a British Uber driver's insurance works; I checked my own insurance policy earlier (exciting times, no?) and it specifically excludes cover for "journeys for hire or reward". Still, there's no need to have a clean licence to be an Uber driver and you don't (as far as I know) have to have your car inspected to a private-hire or taxi standard.