Anyone who believes that brain death is death (and even those who believe that cardiac death is death) must contend with the Gemara, which seems to establish respiration as the sole criterion for life. The Gemara says, "Everyone agrees that life is manifest through the (breathing of the) nose, as it says, 'in whose nostrils was the breath of the spirit of life'."
Now, the truth is that this Gemara even presents problems for those who accept respiration as death, since (a) Chazal held that cessation of respiration is terminal, but we know that it can be restarted, and (b) you can have someone who is quite definitely dead in terms of both brain and heart, but is still breathing via artificial means - one can even make a corpse breath. Be that as it may, let us discuss the various ways in which those who accept brain death deal with the Gemara's exegesis, and I will show that my discussion this week about Ramban offers an additional answer.
The first approach is that the exegesis only refers to someone who is spontaneously breathing, via instructions from the brain. Otherwise, the breathing is no more significant than that which can be accomplished artificially with a corpse. This is the view of the Chief Rabbinate of Israel, and it is perfectly reasonable.
Another approach is to say that the Passuk is only being used to say that if one is looking for respiration, the nose is the place to look for it, but not that respiration is the sole determinant of life. (This was said by R. Moshe Feinstein.) As an elaboration or variant on this, some say that the exegesis is an asmachta rather than a derashah - it's just a Scriptural hook on which to hang a pre-existing idea, but not a Scriptural source for an idea. (Some would say that even a derashah itself is not authoritative - even Chasam Sofer says that derashos are not Sinaitic - and we see that there were derashos performed to show that the firmament is solid.)
But now we have a new answer. The exegesis from the passuk shows that life is detectable at the nostrils. But what kind of life? The original passuk is talking about animals as well as man, which all died in the deluge: "Everything with the soul of the spirit of life in its nostrils, from everything in the dry land, died" (Gen 7:22). So the breath which is detected in the human nostrils need not necessarily be the breath of the nefesh hamaskeles - it may be the breath of the nefesh habehemah. The presence of the breath of the nefesh habehemah is nearly always a perfect indicator that the nefesh hamaskeles is also present, and certainly functions that way for the case discussed in the Gemara. But in the unusual modern case of a brain-dead person, the presence of a nefesh habehemah, indicated by breathing, does not mean that there is a nefesh hamaskeles is present.
Unfortunately, one really can't prove anything either way about what the Gemara meant. To my mind, it really shows why this Gemara can't be used as a source for determining time of death in modern situations, which are so different from the situations that Chazal dealt with. The RCA document expresses this point very well: "the entire purpose of this סוגיא in יומא is to offer practical direction to those involved in a rescue from a collapsed building, and not to address the deeper issue of what actually marks the end of life." (Unfortunately the RCA quotes various Poskim who do not seem to have appreciated that point.) The deeper issue of what actually marks the end of life has to be figured out by studying traditional views of what human life actually is, not making inferences from a discussion in the Gemara which is speaking about an entirely different topic.