What about people who turn up on a boat (their own or a ferry/cruise ship) or get the tunnel?
Can't do much about those coming alone, but no travel without insurance would seem to be easy to implement; all the infrastructure is already in place.
What about people who are here for a long time?
If you are registered with a GP it's no longer a problem.
My point is that the easiest way to solve the issue, if it is one, is to do it at or before the point of entry; it's an administrative issue, not a healthcare one.