The suggestion I made doesn't need extra enforcing

Licence becomes ID card, which includes third party cover. Not driving or passing a test doesn't preclude you from having one. In fact if issued at 16, then you have to present a cycling proficiency certificate with the application, which must be submitted prior to 15years ten months. Then on your sixteenth birth day you receive your id card, complete with cycling licence. At 17 you add provisional driving to it and so on. Insurance could be paid as part of normal deductions, say £10 per week. This provides all legitimate road users third party insurance for ever. Anyone driving without a licence is automatically uninsured, but any third parties would be covered same as currently. If you want extra insurance cover, then you simply pay to top it up. Car tax gets added to fuel, say 10p per litre, that way everyone pays that, again no avoiding it.
Anyone found in the event of an accident not to have a valid ID card with appropriate entitlement, faces prosecution. Cards could be renewed every five years, and bans/convictions would require a new card to be issued. Simples
