Where is the harm in letting her bare the children of her dying partner?
Hypothetically, let's say that she does obtain the sperm, has the child, has the child recognized as the man's child, and then the man dies.
Assuming the man either has no will or has a will that provides for his children generally rather than specifically enumerating his children (saying "to all my children, share and share alike," rather than "to John..., to Mary...," with expressly no allowance for unnamed children) then the unborn child rapidly complicates the probate of the will. Under such a circumstance, people would be harmed by the unborn child. Those people who may be harmed are unknown to us though. They might be other children of the man, they might the close relatives (siblings, parents) who would be in line to inherit if there were no children, it might even reduce the inheritance of the common law wife herself.
If there are more complicated wills and trusts involved, where someone is the beneficiary of a trust that passes to children upon the beneficiary's death (assuming that would evade the Rule Against Perpetuities) then things can become even more complicated.
Examining the issue strictly from an inheritance standpoint and assuming the unborn child would be recognized as the man's child for the purposes of inheritance then someone would assuredly be harmed by the birth of the child. The recipient of the harm and the extent of it is unknown to us though.
I don't see the problem. The whole point of this issue is that the guy is not aware of his surroundings anymore. What does he care what happens to his sperm? He doesn't, because he simply can't.
Law may say otherwise, but law can be silly like that.
By that logic, you might as well say that wills shouldn't be enforced because people will not care what becomes of their estates after death. If that's your view, then that's fine but it runs up against the common view that people should have a say in what happens to their belongings and their bodies after death.
Now things would be different if there was actual reason to believe the guy wouldn't have liked it. He of course still wouldn't care at this point, but he at least would have cared when conscious. But as it is we don't seem to have reason to believe one way or the other, and there is no reason to respect wishes which are entirely imaginary.
I do not agree with the premise that it should be okay to do whatever you want with a guy's body in the absence of his express statements to the contrary. I should not have to say that I do not want my dead body to be thrown to a pack of hungry dogs and my heirs should not be able to argue that they should be permitted to throw my body to hungry dogs by arguing that I said I never wanted that done.
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Relatively few people have mentioned that in the story the man's condition has stabilized and that he is no longer on a DNR order. This makes it sound like it is possible (although perhaps unlikely) that he could recover. Does this change anyone's analysis? How would you feel if your partner took your sperm, engaged in artificial fertilization, and was pregnant after you recovered from a lengthy coma?