My GP surgeryhas instituted a ( another) new system. Initially I was sceptical, but a couple of years in I have to say it works.
The proces starts by me phoning the surgery. OK, it can take a few attempts to get through, but there is an emergency option if the call is urgent.
An administrator then fixes a time that morning when a GP will ring and triage me. From this, I could be given an urgent consultation booking, one further ahead if it’s less urgent, or a n appointment with a Nurse Practitioner, or a Pharmacist, or another professional as indicated from the triage. It’s also possible to book specific appointments with, e.g. the Pharmacist. He is really useful as many appointments are to review medication.
The system seems to make best use of GP time, and ensures that I see the person who is best placed to treat me.