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.
the same thing goes for physical labor. Exhaling is mechanically how you lose body fat. Breathing more isn't sustainable without creating demand, but if it were sustainable you could just breath faster to lose weight.
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.
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.
You miss the point of the parent, which is that exhaling is not a chemical process. The point being that the CO2 you exhale is "somehow produced by something", and once it's there you have to exhale it.
That's a way of saying that breathing faster won't burn more fat just because of the breathing: burning fat is a bit more elaborate than that.