There’s evidence the Mediterranean diet improves depression… I have a feeling Deakin University ran a study on it a few years ago, but don’t quote me. (Remembering that one study is typically only hypothesis generating, not practice changing.) Everything else is “not evidence based” - basically, diet research is really hard, it’s impossible to blind people so placebo effects are uncontrolled… and there’s virtually no research out there anyway, because… big pharma isn’t funding it. But… placebo effects are significant, so if it’s not too onerous, it can be worth trying an intervention just for the placebo benefit.
Incidentally, I was in Central Europe recently, and thought it was a child’s dream - I was amused at the absence of veggies (apart from potatoes) on the menu. How we go on and on about the importance of vegetables here, yet they seem to manage just fine without them LoL and don’t have an epidemic of psychiatric illnesses as a result either. Made me 1) hang out for a few veggies, and 2) think we stress way too much about the correct diet.
I remember Putnam’s truism - twinkies (sweets) with friends is healthier than broccoli alone - that social support/friends is more important than a good diet, to living a long life. (Evidence shows this.) Doesn’t mean we can’t do both, but there’s a limit to what diet will do.
So basically if it’s not costing you anything much, and you want to try pre/probiotics, go for it! I’d recommend kefir or some other yummy way of having them, and see if it helps. But as to the mechanism, or confounds…