The real training is objectivity, which is very difficult for even doctors to maintain, despite knowing just how misleading research results can be and so how they must be wary, how to rationally follow the literature, etc.
I seriously doubt more than single-digit percentages of people googling things, even intelligently, are going to be following citations and weighing competing models. Likely less than a fraction of one percent. And, unless they have personally done similar research and so can understand the data and potential pitfalls more clearly, even that enlightened fraction is highly likely to make some biased misinterpretation.
How can they not, when it is their own health they are investigating? Even a genius layperson examining the literature flawlessly would benefit from an interpretation provided by a neutral third party.
Counterpoint: Doctors have an average IQ of 115, and train across a huge variety of disciplines; mostly while sleep-deprived. Many doctors' training was decades ago.
Anyone with a somewhat abnormal condition, access to Google Scholar, and able to learn how to follow citations and simultaneously weigh competing models can outdo anybody except a dedicated specialist.
The relevant job comparison isn't to a painter, it's to a programmer with a hundred clients, each of whom he talks with for 30 minutes, once a year. Sure, the programmer has much more knowledge and skill--but is he really going to be able to adequately replace the every client's home-brewed solution spread across 4 excel sheets, 3 batch files, and an archived website?
Should the client really be restricted from downloading Python or Visual Studio for their own use, unless a real programmer signs off on the requirement and supervises them?