Although I'm not an expert in the history of UK taxis, my impression is that a good number of the regulations were imposed by the government, due to complaints from the public about quality/maintenance/knowledgeability/numbers. That seems to have been the impetus for the original 1662 Act imposing a mandatory licensing requirement, at least: Londoners thought there were too many taxis clogging up the roads, and further were worried about reckless and otherwise un-vetted drivers.
Even nowadays the requirement that London taxi drivers know how to get around London seems popular with the public; British tourists regularly complain about how drivers elsewhere can't even find major landmarks without a map, in a "glad it's not like that back home" sort of way. My guess is that the requirement would be confirmed if put to a referendum. Other requirements, such as the "cannot refuse a fare in central London" requirement, are also pretty popular.
Sorry, I guess I should say I wasn't speaking with any experience of taxis in London. My statement mostly applies to things I've read/heard about taxis in the US and Canada.
The perverse way that taxis are regulated in the US (medallions and so on) seems to be completely unique to the US, or perhaps North America. It's not how things work in Europe. Uber and friends were started in the US as a sensible reaction against that perversity. They make a lot less sense in Europe.
See also the Coin wallet, various payment services, etc, which were conceived to address inefficiencies that are unique parts of the US financial service industry.
Even nowadays the requirement that London taxi drivers know how to get around London seems popular with the public; British tourists regularly complain about how drivers elsewhere can't even find major landmarks without a map, in a "glad it's not like that back home" sort of way. My guess is that the requirement would be confirmed if put to a referendum. Other requirements, such as the "cannot refuse a fare in central London" requirement, are also pretty popular.