I'll muddy the waters further here because well, science...

Some recent studies have shown that the keto diet has been successful in supporting patients with schizophrenia.

The Mediterranean diet has been shown to be helpful for those suffering with depression. I haven't seen studies on the Paleo diet as it relates to mental health, but I imagine the Paleo diet would also assist depression for similar reasons. I couldn't find any Paleo mental health studies, there is quite a bit of conjecture about it.
Six Reasons to Go Paleo for Mental Health | Psychology Today

All three diets are potentially high in omega 3s depending on the proteins you select. That seems to be helpful regardless and omega 3s also help to reduce inflamation within the body. This reduction is good for your physical health and your mental health. This is why so many of us are on fish oil.
Https://paleoleap.com/brain-on-paleo/

All three diets are low in sugar and processed foods. In my opinion this is the number one source of benefit to these diets. Sugar and processed foods cause inflammation and mood swings. They are also much more difficult to process. Removing them gives the body a much needed break. This helps the kidneys, the liver, the heart the brain, the digestive system and I am sure it assists in other ways. Oh yeah, healthier skin

Paleo and Mediterranean diets can be executed in such a way that they cause ketosis, so you can do Paleo keto or Mediterranean keto. A Mediterranean keto friendly diet would drop the whole grains and still allow for some dairy while the Paleo elimiates dairy altogether.
5 Things Paleo Eaters Should Know About the Mediterranean Diet

I also found several studies that showed probiotics assist fatty liver disease. The sources used in this study came from controlled probiotics, but there is a lot of information showing fermented foods help as well.
Randomized, Double-blind, Placebo-controlled Study of a Multispecies Probiotic Mixture in Nonalcoholic Fatty Liver Disease | Scientific Reports
Fermented foods are powerful helpers. They are already broken down to an extent, so it is less work on the body to process them. Additionally they contain tons of helpful probiotics. Bonus! They also assist in the reduction of inflammation.

Just some extra food for thought. Enjoy!