Re: Not only their problem - ?
You can easily look this up.
https://www.gov.uk/government/publications/dvla-data-sharing-strategy
The DVLA has a positive duty to _disclose_ vehicle registration details to anyone who makes a reasonable request. It is not private date, it's a public registry. There's no loophole, the system is working as we intend it to work.
Next you'll be complaining the electoral role is also public data.