Adam,
We do almost exactly the same except if a patient says they cannot get to the office to get a script we ask them how are they going to get to the pharmacy to pick it up? (Only a few of the local pharmacies still deliver here). If they say someone will pick it up for them later in the day or tomorrow, we reply, "but if you are that ill to require a script, you really need to start the medicine immediately". 9/10 times, they make an appointment that day to be seen.