When the disease is something that requires frequent reapplication of antibiotics, it's important to confirm periodically that they're working (and that the problem isn't a different bacteria that's amoxicillin-resistant).
It seems to be this is more for the benefit of the healthcare system financially rather than the patient or the health of the population as a whole.
I think that's really where one needs to look to square the circle. The drugs available prescription-free have lower risk of injury or disruption when abused. Antibiotics are not in that category, which is why they require doctor oversight and a prescription. Take too much, and you can damage your digestive tract. Take too little, or the wrong kind, and you can worsen the infection by killing the bacteria that the immune system can control, leaving their food supply for the ones the immune system is struggling against.
They could require less doctor oversight perhaps, but probably not zero.
Amoxicillin isn't available without a prescription because using the wrong antibiotic on a disease can make the patient worse.