The reason that competition freedivers don't die as often as they have blackouts is because they have numerous support team members diving with them that are specifically there to rescue them if they lose consciousness.
What may not be entirely clear from the article is that competitive freedivers hyperventilate. Hyperventilation clears the lungs of CO2 and thus delays the breathing reflex that causes the feeling of suffocation, though without significantly increasing the amount of oxygen. Hyperventilating before a dive increases the risk that oxygen will run out before the breathing reflex kicks in. The training that freedivers undergo is to learn their limits and know exactly how long they can last before this happens.
In the text, there's an oblique reference to how Alexey hyperventilates, in the sentence "The breaths fill his enormous lungs" followed later by "Alexey fills his lungs with one great inhalation". "The breaths", plural, because he's taking multiple, deep breaths to clear his lungs of CO2. Then one last breath to push in as much air as he can.
We then read how he keeps checking the time. A major reason for this, left as subtext to the knowing, is that checking the time is how he keeps track of how much oxygen he has left. Since he's suppressed his breathing reflex he has no other indication of how close he is to suffocating.
As should be obvious, the least error here can be fatal. A blackout is lights out, everybody.
Source: I freedive recreationally, without hyperventilating. I love freediving. Can't freedive dead.
That is exact opposite of what I would intuitively guess. I would guess that it is way more likely that blackout means you body is getting injured rather then opposite.
It's an evolved response the body does to conserve oxygen use in the brain. Consciousness shuts off, but the brain is still active. For example, someone who blacks out this way will not get water in their lungs--a part of the brain that is still active keeps your mouth shut so that no water is accidentally inhaled.
Freedivers try to avoid blackouts, but in this context, there are established safety procedures and it is not nearly as bad as you may think.
https://en.wikipedia.org/wiki/Laryngospasm
Before laryngospasm occurs water must enter the larynx. At which point, if you 're a few meters under the surface of the sea, you're in the process of drowing.
Think of it this way: if it were safe to lose consciousness underwater, if we had a magickal reflex response that kept us safe- why would anyone ever have to drown?
It goes against common intuition, but it's not like that's unheard of when it comes to the human body. Look at intermittent fasting for example.
The safety protocols have also evolved over times. Nicks death for example has triggered a rewrite of the rules that would require a medical examination after a blackout or if you cough up blood. This examination can result in a ban from further competing for the next days for example.
Of course it's not a safe sport, it's an extreme sport in an extreme environment. But lots has been learned in the past decades, as it is still a rather "new" sport and if exercised correctly it is rather safe. Also the article just talks about a competition. Most freediving happens either in training or when people do it recreationally. While accidents happen there as well, these are mostly about the lack of safety divers, that's why the first rule of freediving is, never dive alone.
Finally, on your last paragraph, freediving does change your day to day life, at least from my experience. I just do it recreationally, so I'm far from these depths and times Alexey and the others achieve, but the mental control you learn can be applied in many situations.