Sounds reasonable, even sensible, but, I wasn’t aware we were discussing insurance, I thought we were discussing identification of “vehicle” and driver/rider, in the event of an incident, which may not necessarily be an insurance issue. (Eg obstruction of pavement).
It would seem more likely that a hire escooter would be abandoned on the pavement, rather than docked, than a privately owned one where the owner might be more careful with it. As with conventional road vehicles, it is the law abiding owners who would get penalised if these became road legal, as they would be more likely to have some form of registration, licensing, and insurance so being traceable while the cheats and blaggers who already don't comply with conventional vehicles' legal requirements would just apply their philosophy to escooters.
While it seemed a wizard wheeze at the time to start trial hire schemes it must have been obvious that members of the public having seen them about and even tried them might have also seen them on sale on the internet and might have fancied one for themselves. With loose enforcement and apparent poor information on the restrictions applying to them, there are illegal private ones about so one can expect that if private ones were to become road legal many more illegally ridden ones would appear using the increase in numbers of legal ones as camouflage due to safety in numbers. Some kind of provisional system for identifying and regulating the sale could have been considered, eg registered dealers only, with the same requirements as hirers for sight of some kind of driving licence and insurance before being allowed to take it away. Presumably you can buy a car without any of this, even online, so maybe that ship has sailed already.
Short of electronically tagging the entire population, some kind of electronic marker, number plate, insurance and registration and the requirement to wear a BSI marked motorcycle -standard helmet (it's technically a motor vehicle) ought to bring home the concept that it's a vehicle, not a plaything. Similarly, the same powers to confiscate and crush your vehicle together with a hefty fine and points on your licence if it's not insured ought to act as some incentive to comply. The likelihood of this happening needs to be real enough, even if relatively unlikely, that potential buyers will take it seriously.
Maybe that all sounds a bit OTT, but either it's a motor vehicle, or it's not, it's either something that can only be used as part of a hire scheme or it's available to anyone to own as long as they comply with the rules, or it's not. Too much woolly thinking has led to the current situation, and it's possible now that the toothpaste is out of the tube it will be difficult if not impossible to get it back in. Just because most people are law abiding (if it's clear to them what it is) it doesn't stop the minority who aren't from carrying on as they like.
Whether the DVLA has the capacity to deal with the extra vehicles or not? Highly likely that it does, as it's just an expansion of the existing system, if escooters are to be seen as just another category of vehicle. Presumably the existing hire scooters are already somewhere on their database, with the registered keepers being the hire companies.
So, if the grey areas were to be made clearly black or white, if the police were to be given enough resources to exercise the powers they have already got, if anyone really cares enough to sort out the potential mess, maybe stop being so Anglocentric and see how other countries deal with the issues maybe there will be a clear cut solution to the situation.