Buteyko practioners build up the ability to work very hard while only nasal breathing over the long term. The point is to learn to modulate breathing in a way that keeps a certain kind of blood chemistry (CO2 levels) and cellular oxygenation.

If you commit to nasal breathing for exercise as a constraint, it does force you to modulate your exertion while also increasing your CO2 and developing, according to the Buteyko folks, a new baseline for respiratory health and capability.

If you're running for your life from a tsunami, by all means mouth breathe. If your purpose is maximum exertion, of course mouth breathe. But that's not the only possible purpose of exercise. It can also be about respiratory training. Nasal breathing becomes a natural guideline/modulator for long term improvement in that regard.

A few counter anecdotes: I read a story about a baseball coach making the players hold water in their mouths while training to make sure they couldn't mouth breathe. I'm pretty sure if you watch an MMA fight the fighters will mostly have their mouths closed. And the airway-focused orthodontist Bill Hang ran a marathon with tape over his mouth (only removing it for hydration) to make a point about the importance of nasal breathing. Also, I can do sustained moderate to high intensity exercise with only nose breathing.

Nasal breathing has a lot of benefits and it's arguable that maintaining your level of exertion such that you're always breathing nasally is a good technique for building cardiovascular & aerobic capacity.

I’m actually good friends with Jacob Kiplimo. He swears that the reason he was able to have such success in the half last year was by exclusively breathing through his nose during training.

My one HUGE gripe about Buteyko in particular is the nasal breathing bit. Breathing through your nose is good imo, but I don’t think it needs to be such an important goal all the time