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Jayciedubb It is interesting as a "foreigner" to read posts like yours, to compare or contrast different countries' systems.
The need for a car is widespread in much of the UK and likely many other countries, especially in rural areas and even some suburbs where public transport is scant or non-existent. Even many trading-estates are not served directly by buses, though there may be stops on the main road passing the estate.
A vicious circle sets where public services diminish so more people need cars, so the buses or trains lose trade, so....
In some areas here the railways have gained over the last few decades as long-distance - many >100 miles - commuters with city-based work drive to the nearest station and continue to town by train. Most still need the car between home and railway-station though.
The registraion laws in the UK seem fairly similar to yours, with the mutual dependence of the tax, insurance and Mot (annual roadworthiness inspection) status.
Though the Driver & Vehicle Licensing Agency (DVLA) holds a record of all vehicles, "registered keepers" and legality, so the Police can easily verify these even before stopping the car, and they do not need the driver to present any documents there and then.
The public has limited access to the database to verify tax and MoT status of a car offered for sale, or to report ones apparently abandoned, but DVLA keeps the owner's identity confidential on the public pages (probably protected by the Official Secrets Act.) DVLA's web-site, and many other "dot-gov-dot-uk" ones, are very clear and simple to use.
I think one major difference between the US and UK is in how every motor-vehicle is registered for public roads.
Each British car, motor-bike, lorry, etc. has a Vehicle Identification Number on a plate in some obscure place, and the registration-number on the displayed plates. Both numbers are intended to last for the whole life of the vehicle irrespective of owner and any use on public roads - the number-plates are never changed.
The system is conformable across all vehicle types, irrespective of owner, for any use on public roads: from an exotic Italian sports-car to a moped, a bus or lorry, road-repairs or agricultural machine to Army battle-tank. Even a miniature traction-engine though only by its owner applying to be allowed to drive it on public roads.
The registration-number can be traded, to a point, via DVLA, so an owner can have a particular registration-number transferred to a new vehicle, but it is expensive and not really encouraged. I know two people who have done this.
Indeed, I was once pulled in for the MoT test having expired. I had not realised this, but once stopped and told the problem, I twigged I had muddled the MoT and Insurance expiry dates. The Police Officers accepted this was a genuine mistake and let me go with a warning to complete that journey but not use the car again until taking it for a booked test.