Restricted shitty breathing, both chronic and short term, jack up your blood pressure, stress and cortisol levels, etc. Healthy athletes know how to breathe. Chess is no different. You can be a great chess player in spite of your terrible body mechanics, breathing patterns, etc. It doesn't mean it's "good" for you, and certainly doesn't mean it's unavoidable.

Exhaling is how you lose fat, because you're sitting on your chair all day. That's the baseline. If you were actually doing sport you'd lose fat by breaking down ATP when repeatedly contracting your muscles.

As a science puzzle, would one feel hungrier sooner if one intentionally decided to breath faster for a few hours and kept at it?

I don't think that's how you lose fat. You get rid of the exhaust generated during the chemical fat burning process. If you couldn't exhale you would still burn fat.