No one has been working on the USDS for a long time. It was created in 2014 to _improve archaic systems_. That was its mandate, and it came from the top.
Elon's not doing anything special or respectable. Decisive action and moronic flailing are not mutually exclusive.
And I'm not an expert, and maybe you're not in England, but NHS England has supported online appointment booking for about a decade. Can't imagine it's that different elsewhere in the UK based on a quick search.
https://www.england.nhs.uk/gp-online-services/about-the-prog...
Think stripping out the copper wiring rather than doing a remodeling.
Obviously someone did since there was "Choose and Book"[0] which was supposed to be live in 2005[1]. But, as anyone could have predicted, farming the contract out to consultants did not (and has continued to not!) work.
Also pharmacists were calling for themselves being able to electronically book GP appointments for (patients? customers?)[2]
An ICO report[3] also implies there were plans for online appointment bookings in 2006.
[0] Ok, not strictly GP appointments but NHS appointments nonetheless.
[1] https://www.digitalhealth.net/2005/01/e-booking-to-miss-its-...
[2] https://www.digitalhealth.net/2005/12/pharmacists-call-for-g...
[3] https://ico.org.uk/media/about-the-ico/documents/1042390/sur...
You asked yourself how would the system work best for you. Not how would the system work best for everyone. Elderly people make up a large proportion of GP appointments. 20 years ago they weren’t going to be booking anything online. Even today there are many that won’t.
Elderly people can still book over the phone. Use your brain.