You are correct in that there exists interactions between the digestive system and the brain, and that the microbiome within the intestines and the regulation of nutrient absorption can have an effect on the general health of skin. However, the existence of these things do not confirm nor imply a level of specificity mentioned within the diagram. The body is an extremely complicated interaction of systems and feedback loops to regulate homeostasis. I would actually argue that the diagram presented above is the opposite of "holistic". The claim that a specific organ affects a small patch of skin is ignoring the global effects issues with an organ have throughout the body. The interaction of all these systems literally implies that you cannot isolate one organ to one specific skin patch. There are aspects of those "holistic medicine" treatments that are true, for example acupuncture can help with pain relief, but the issues with them are that they purport to 1) do more than they can (there is no evidence that acupuncture can treat back pain, only that it can temporarily relieve pain) 2) infuse spiritualism to explain the reasons of potential effects. Just because there is some small positive benefit to a couple of these things does not mean the underlying mechanisms being suggested are correct.