As a future medical professional I see this as absurd. Look around the news and see the rampart issue of counterfeit drugs then think again about where you get your medication from. Not only that, but uses like this is what is contributing to the rapid development of resistant bacteria. Drug companies aren't putting as much effort into researching antibiotics like they used to, and if I recall correctly we're several years off from new antibiotics. I highly doubt your average layperson knows which gram negative and which gram positive variants amoxicillin exerts its bactericidal effect on or how to identify if that is what they are suffering from, what the common side effects are and how to identify a type II hypersensitivity reaction.
Any time I have ever gone to the doctor with an infection, I have never had to wait around for the doctor to do a Gram stain or culture or anything to determine exactly what I had, nor did the doctor make any effort to follow up and make sure I completed the course of treatment. So what difference would it make had I gone and just self-medicated? Having a prescription label did pretty much nothing to solve the problems you bring up. Sometimes the advice of a doctor is invaluable, and I will be glad to go and get it. Other times, I just need more cream for psoriasis or whatever, and it's exactly the same psoriasis I had last time I saw the doctor, and it's a waste of time and resources to go get another prescription that he's going to give on the spot anyway just when I tell him "yeah, it still works, I just need more". I may as well just order it online.