It is a natural question, and a reasonable one: if speed matters so much, and the race against cellular decay begins the instant the heart stops, why not start earlier? Why not preserve someone while they are still alive, before any ischemic damage occurs at all? There are two reasons, and they reinforce each other. One is legal. The other is biological, and on its own it would be decisive.
The legal reason
Cryopreservation can only begin after legal death has been pronounced. Beginning the procedure on a living person would be homicide, full stop. This is not a bureaucratic technicality that a clever provider could route around; it is the bright line that separates medicine from killing, and a serious organization does not go near it. The same constraint shows up as the legal-death paradox in the race against decay: the ideal biological moment to start is one the law rightly forbids us to use.
The biological reason
Even setting the law aside, the procedure itself would kill a living person. To achieve vitrification, most of the water in the body is replaced with cryoprotective agents. At warm, living temperatures, those agents are toxic. That is why the agents are perfused cold, into a body already chilled to near 0°C, which slows their chemistry along with everything else. That temperature is itself incompatible with a beating heart, so the state that makes cryoprotectant perfusion workable is one no living person is in. They are safe only at cryogenic temperatures, where all chemistry, including their toxicity, is paused. So perfusing a living body with them, and then cooling it to glass, is not a survivable process by any current means. It is the very thing that makes preservation possible, and also the thing that makes it incompatible with being alive.
This is also why rewarming is one of the central unsolved problems of the field. The same toxicity that rules out preserving the living would have to be cleanly reversed on the way back up, and doing that is part of why revival is currently not possible. Researchers are working to reduce and eventually neutralize cryoprotectant toxicity, but that is a long road, not a solved problem.
You cannot be cryopreserved alive for two reasons at once: the law calls it homicide, and the biology calls it lethal. Preservation is a procedure for after legal death, not a substitute for staying alive.
Perhaps, in some distant future where revival and gentle rewarming are routine, preserving a living patient might become a real medical option. Until that technology exists, though, attempting it would simply mean certain death rather than a preserved chance. For now, the honest position is the careful one: we wait for legal death, then move as fast as the clock allows.