I'm not sure how being a parent provides qualification in assessing the practicality of this startup, but if it does, then I would like to weigh in on this issue as I am a parent of 2. These clinics are typically staffed at least by a nurse practitioner, who can treat and prescribe medications (depending on the state) for almost any reasonable non-urgent condition, including antibiotics. If you live in a state where the nurse practitioner can not prescribe medications, there is usually a physician on staff that works along side the NP to assist in those situations.
These clinics sell themselves as being non-urgent care facilities that are able to treat conditions that otherwise would have required a physician or Emergency room visit. If they could not prescribe medications and treat most of these conditions, I am unsure how successful they would really be.