So you propose a separate, isolated network linking all the medical facilities, doctor's offices and private practices nationwide? Even the military doesn't do that for most of their offices.
Also, the doctor's computer pretty much needs to interface with the system(s) that handles patient billing (and thus non-medical companies) and the system(s) that handle patient scheduling, reminders, etc.
Not really an issue in the NHS, apart from the occasional non-resident foreign national.
(The "fundholding" system does mean there's a certain amount of internal billing which the patient is never aware of, but the beating Bevinist heart of the free-at-point-of-use system is still in place)
Free-at-point-of use process tend to be ones that require integration with a billing service, namely, to send information about the performed procedures to whatever system is paying for them, no matter if it's some state agency, private insurance, or whatever else - that's what I meant by non-medical companies that would need to be on the network.
A private practice where everything is paid by the patient in full by cash or CC could do without any integration with external systems (just run a standard cash register), but as soon as someone else is paying for it, you generally need to link the doctor's office systems to that in some way.
Also, the doctor's computer pretty much needs to interface with the system(s) that handles patient billing (and thus non-medical companies) and the system(s) that handle patient scheduling, reminders, etc.