Centuries of study show us that many homeopathic cures do work.
I am not sure that you understand what "study" means in a modern scientific sense. If there are any of those showing the efficacy of homeopathy, please provide links - I am genuinely interested in seeing them (not snide, seriously).
As an example, I have a medical doctor who suggested drinking camomile tea to help me sleep, and it works. He could have prescribed a man made chemical to do the same thing
Now I am not sure that you understand what homeopathy means. Read the linked Wiki page to understand (it involves ingesting ridiculously diluted chemicals to purportedly cure illnesses on the utterly unsubstantiated theories of "like cures like" and "water memory."
I think what you're referring to is naturopathy, which is a whole different kettle of fish. My wife is a big believer in naturopathy, and while I think some of it is touchy-feely new age quackery, there is no dispute that naturally occurring plants, herbs and other medicinal sources can be effective healing tools. So no argument there.
My personal $.02 is that many people who prefer naturopathic medicine and oppose GMOs - my wife among them - do so not from a scientific viewpoint but from a moral viewpoint. Many of us would much rather trust things that grow naturally than are made artificially. But while it may ultimately prove true that some GMOs are harmful, I strongly believe that we should come to that conclusion through scientific study, not because we "feel" that something lab-created is inferior to something made by human science.
Nature made the Black Plague, tobacco, lard and the Destroying Angel mushrooms, too. Just because it's natural doesn't necessarily make it better for you, or make man-created things bad.