the diving reflex is real, and is used clinically to treat supraventricular tachycardia (where a re-entrant electrical feedback loop in the heart can be broken by slowing down the choke-point through which it passes). It slows down the heart and lessens its contractile power consumption. That could conceivably work (in my mind) after a cardiac arrest, but it isn't really related to brain death (which is an issue in the days and weeks following, in the ICU where oxygenation and glucose usually plentiful).

Remember: the "theory" behind any medical idea is almost irrelevant if you have data regarding the outcome. If you have outcome data, it supercedes the theory, and is all-important. The theory simply helps us prioritize which treatments to assess first - the better the fundamentals, the more worthy it is of scarce research resources.

That's one of my biggest criticisms of the lay press: they emphasize the scientific narrative and don't focus enough on the outcomes. The most meaningful outcomes in scientific terms always mean the same thing: a meaningful improvement by blinded observer in something we care about.

"meaningful improvement" - this means that improving survival by a tenth of one percent might not be worth the trouble of implementation

"blinded observer" - because it cancels out the placebo effect and expectations

"something we care about" - such as returning to school. But if you do a SPECT scan and the SPECT scan "looks better" to the doctor? That's not important. We sometimes use things that are well-documented to reflect important things even though they are not important in their own right... we call these biomarkers.

I have no idea about the treatment you describe, but the mammalian dive reflex is very real, and is a useful way to induce bradycardia. For example, splashing cold water on your face can legitimately help with a panic attack