I think the reality is closer to different dysbiotic states can cause depression. Depending on the the specific dysbiosis, the interventions will be different.
I'm sure you could - but if you went about it scientifically there would easily be thousands (not just the 3 in Ayurveda). That being said, I do think Ayurveda is mostly about manipulating the microbiome, and I think there are probably real nuggets of truth in its approaches. For example the use of Triphala to improve motility will help in numerous dysbiotic states.