As you wrote, diet is one contributing factor only. If other factors are in balance, one off kilter may not push overall mental health down.

What are some attributes your dad has that help him function well? I'm sure there are many.

Some people obsess about weight and some simply don't. Some people stress about being widely liked and others don't. And other examples. It may come down to attributes like self confidence/esteem, feeling good in one's skin, mental fortitude/psychological resilience, etc. We all also have our own differing priorities in life.

I bet there are many people here that are my height and weight who regard themselves as distressingly overweight. I don't. I think I look great. Perspectives vary. I look in the mirror and see a hot chick. My brain sees it that way. However, I have other things in life that I'm a perfectionist about, but none have anything to do with physical appearance. I also generally see the glass half full, not half empty. While some here post about distress every day, I tend not to that much. However, sometimes I deny or stuff stress. That has its good aspects and also bad ones.

I imagine it's extremely difficult for most to change perspectives and other attributes from youth and later, but not impossible. Nature and nurture.

There's also sometimes the need for acceptance that leads to coming to peace with something. This also applies to most grieving processes and helps us move on rather than be figuratively jailed, long-term. This is a type of choice to make, but again, isn't always easy.