As emergency medical workers know, you're not dead until you're warm and dead.
It's not practical to confirm brain death in hypothermic patients; Some apparently dead people with severe hypothermia come good when you warm them up - particularly if they had a high blood alcohol level to begin with.
But essentially the OP question is ass-backwards. Death is defined as a state from which the brain cannot be restarted, so if you can restart the brain, then you were (by definition) incorrect to believe that patient was dead.
That's a rather flexible definition of death. Suppose you could slice a brain up and scan the slices with an instrument precise enough to record every neuron and their connections, and then later simulate those neurons in a computer. Clearly, the original brain is dead. But the brain function, which is really what it's all about, would be replicated and restarted in a simulation. So by your definition, the sliced-up brain would not in fact be dead, which is a bit counter-intuitive.
What about those cryogenically frozen heads then? We don't know whether the brain cells are undamaged enough to be scanned sometime in the future, nor whether anyone would have an economic incentive to do so, but that is at least hypothetically possible. Would it be a misnomer to call these people dead regardless?
Medicine isn't a logic puzzle. Almost all of the definitions in medicine (and for that matter, many of those in biology) are woolly and vague around the edges.
If you are brought into a modern first world ER with a body temperature of a few °C, and no apparent signs of life, but equally no signs of trauma incompatible with life, then they will attempt to revive you by carefully increasing your body temperature. If you don't exhibit any brain activity once your body temperature is over about 30°C, then they will (retrospectively) declare that you were dead on arrival.
If you are brought in with your head detached from your body, they won't hesitate to declare you dead.
Nobody's going to say "Stop! You can't issue a death certificate until you are confident that this person's brain couldn't be scanned and replicated in an operational state at some future time!". But maybe one day, they will.
Definitions of death have changed as medicine has improved. Not so long ago, death was defined by the absence of cardiac activity, rather than the absence of brain activity - indeed, I believe that's still the legal standard in Japan, and it's a cultural belief common in South East Asia. This leads to all kinds of problems for patients on heart-lung machines, and for dead people whose heartbeat is maintained by a pacemaker, for example - the former can be declared dead, but then recover, and the latter can tie up significant resources looking after a corpse.
Human life is poorly defined, but then "life" is too. No good definition of life exists - there are lots of definitions, but none include all of the things that people want to include, while excluding all the things that people don't want to consider 'alive'. Typically they are a grab-bag of features, not all of which need to be present. For example, the ability to reproduce appears in many definitions, but as this implies that a vasectomy is an invariably fatal procedure, it clearly isn't necessary; And as some crystal structures reproduce, but are generally excluded from consideration as 'alive', it's also clearly not sufficient.
I strongly suspect that the real problem is that 'life' isn't a thing. There's no real dichotomy of entities with all falling into either "alive" or "not alive", and the only real reason why this false dichotomy is so popular is a hangover from dualism - itself a hangover from religion.
"Alive" is what we call our preferred collections of complex cyclic chemistry. The 'preferred' and 'complex' terms render it a matter of pure opinion.
Is a virus alive? It is if you want it to be.
