There are a lot of problems with the NHS. The GP booking systems are diabolical at the moment, as you point out, and GP receptionists are up there with post office staff for aggressive service. I personally use Babylon GP at Hand and am very satisfied, but that's only available in London I think.
That said the reason we don't do as much in the way of screening and yearly physicals of young healthy individuals is because there's very limited evidence that it's even helpful and sometimes even evidence that it's harmful.
That isn't to say they don't sometimes save a life, but they also cause unnecessary medical procedures and needless medical anxiety. What kind of early cancer detection is really happening at a yearly physical? You could argue that a doctor telling you to lose weight might help reduce obesity or diabetes, but so far that doesn't seem to be working.
Even if you do catch and treat a cancer, was it a slow progressing cancer that never would have caused an issue for the patient? This is particularly significant when you're looking at someone in their 80s.
If you have any health conditions, have a family history of disease, are taking repeat prescriptions or are above 40, you will start being offered regular checkups and screening.
You can also book a private GP or checkup if you want, or even just pay for a periodic blood test. The NHS/NICE has decided the evidence base doesn't exist to support it, but you're free to decide otherwise if you wish.