Hi,
sorry, I think you got it all mixed up:

Emergency Clinic: you go to Triage - then see the Doctor - then possible referral to Physio.

Private Physio practice: doctors referral and/ or people walk in from the street with a problem and ask you "What is wrong and how can I get better."

At least in NZ and Australia you do not need a doctor's referral to see a Physio and I believe this is the same for the UK.

Your correct, you have good control over your working hours when you run a private practice - you usually work twice as much as anyone else and during hours where everyone else is at home (unless you are on the on-call rota in a hospital )
I have got far more control being an employee.

Cheers, Fyzzio