I think you too are confusing science and non-science at places.

Regarding when life begins, science is extremely clear: unfertilized eggs are alive, and so are sperm, and so is the fertilized zygote. There is no "begin" with regard to life. It's all alive, all the time. The fertilized zygote begins existence as an alive fertilized zygote once it is fertilized. There is no magic spark of life essence that enters at some point. Any reasoning about "when life begins" is hopelessly confused scientifically.

Regarding what counts as a person, science can't tell us, but it can tell us when it is appropriate to reason by analogy, and that's what Emma was completely appropriately doing with brain death. Your counterargument hypothetical is not particularly helpful because science tells us (1) not gonna happen, and (2) if you do get consciousness again, it's not going to be the "same person" but rather effectively their identical twin, who has to develop mentally in an old body. If this kind of thing did happen, I don't think we could predict how society would respond. It therefore is not a very compelling counterargument, though it is a very interesting thought experiment.

In any case, we have people's revealed preferences and explicit preferences (e.g. codified into law) telling us over and over that the key to who someone is is their mental identity. It's backed up by lots of fiction, too. Science can tell us about the existence and continuity of mental identity. So, yes, to an extent deciding that this is going to delimit what we call a person is arbitrary, but there is a powerful argument from consistency that is lacking in the "moment of conception" argument. In this sense, it's not arbitrary; it's just reading out what we already believe in other cases.

Of course, we could decide this case differently. But generally we tend to require decent reasons for making a distinction.

Back when life seemed magical, imagining a moment when the spark of life entered a woman and created a new human was a decent way to delimit when human life begins (and suggests a clear way to delineate human life from other things). But at our current state of knowledge, we realize that was part misconception, part oversimplification.