Note, I gave two versions of the secretary :) - the non-professional and the professional one.
The professional secretary I described is a nurse. Also, the non-professional would effectively be a nurse. They just have different duties where the latter would also check-list through the doctors remedies. Note that this potion is not that of non-autonomous robot. They have the right and duty to double check on the doctors performance.
First, costs. Where I'm from nurses are fairly well trained professionals, but have a really low salary. They seem to keep at it. I could imagine a worse gig than GP:s assistant.
In single payer healthcare one should look at the total cost/benefit for the entire economy, not just how to minimize the cost for the single health center.
The cost of bad diagnosis or faulty recipe can be that the time of the doctor visit was completely wasted. Hence, the time used for the visit by the patient was completely wasted. If they need the medicine they need to reschedule - thus the cost doubles not only for the doctor, but also to the patient. And the direct cost is not the only cost. As we are effectively doubling our resource use, now we are clogging the system, and delaying other patients.
And. I don't get it why this needs to cost more. They have plenty of nurses at the health centers, for one accident or the other. They could just change the routines that the secretary-duty nurse gets a call if she's needed at some other place.
Of course, this is me looking at the situation through the lense of the single payer social healthcare scheme. The logic of the situation is probably different elsewhere.