There are definitely better ways, but people have to be willing to consider change as improvement rather than something to fear. Unfortunately, the current government is not trustworthy and any improvements via privatising areas of healthcare will rightly be viewed with suspicion. We can all imagine them outsourcing a multimillion pound digitisation project to their mates who aren't the best qualified option and it's another ridiculously outdated lame duck before it launches.
I'm in Canada and my province has a Digital ID for all healthcare. It contains a record of all previous test results, shows me test results as soon as they're available, all previous and upcoming appointments, vaccination records, medication history. I once couldn't remember the dosage of iron my daughter takes when I was at an out of town pharmacy - the pharmacist was able to use her ID number to look up what she normally has prescribed and get me the right one without a prescription.
We have a hybrid of private and funded healthcare, in that it's free to the user, but private clinics offer the services and bill the health service for them if the GP referred you. So my GP gives me the request form, and I can call or online book any one of 4 ultrasound providers/blood/urinalysis providers within my town and see who can see me at the most convenient time. I can always be seen within the day if I want to. The provider uploads the results to the digital ID and I still don't pay anything at the point of use, but because they're private businesses it's an entirely different experience. They're competing for my custom and make sure that they have convenient appointments available. These clinics also offer privately insured / paying patients other lucrative services (cosmetic stuff or physio etc) which keep them extremely profitable.
When my daughter needed to see a specialist, I was given their office number by the doc and we made an appointment over the phone. They don't just pick a day and summon us!
I work at a private psych clinic, and most of my clients are funded by private insurance, but I also see referrals from the health service who need my area of specialty. They all receive the same care from my clinic, although to get the referral from the health service, clients are usually in a crisis situation rather than pre-emptively concerned.
I get a text reminder a week or so before an appointment with "Reply C to confirm" and that's the whole appointment reminder unless I login in to have a look in advance. But I made the appointment rather than waited to be granted one, so it's already marked up in the calendar. I am shocked that the NHS are still sending letters through the post and running a "surprise! Here's your appointment time!" system. But also not shocked at all unfortunately...