Being somewhat familiar with how this plays out in the NHS, all I can say to this is: that would be very nice. And that's with a system that has an ID number which is legislated to be the primary personal identifier.
Substantially all touch points with the NHS that don't flow through NHS Login require the NHS Number plus two other bits of demographic information, specifically because the interconnected databases can't be relied on to have either one number per person (for which there are valid reasons, but never more than one active), or one person per number (for which some DBA somewhere needs to answer for their crimes against relational consistency).
There is a non-trivial number of medical procedures (some remarkably serious) which are performed on the wrong person every year because they were mis-identified at check-in, or later. Solving this is not as easy as you would think.
What you actually want is better control over how government agencies handle your personal information, but I guess people are less spooked by ambiguous worded legislation and databases than by physical pieces of plastic.