Appointments and the token queue
Appointments, walk-ins and a live token queue per doctor: who is waiting, who is with the doctor and who is done, visible to the whole clinic at once.
The queue is the default screen. Each doctor has a column, each patient a token, and the status is the colour: booked, arrived, waiting, with the doctor, done, no-show. Changing a status is one click and the front desk and the doctor see it at the same moment.
Booked appointments and walk-ins share one queue. A walk-in gets the next token for the doctor they came to see, and an appointment holds its slot, so the front desk is not keeping two lists in their head.
Delays are shown rather than hidden. When a doctor runs behind, the waiting time on the following tokens grows on screen, so the front desk can tell a patient how long they will wait instead of guessing.