This is indeed an intellectually consistent, bright line. It also permits the gruesome procedure of 'partial-birth' abortion right up until the actual moment of birth, and can be construed to allow the killing of the child if it can be prevented from taking a breath after it is born.
There is an even brighter line that can be drawn, and that is the moment of conception. This line has the advantage of being supported by all of our knowledge of biology and medicine.
But even at that point, it's interests do not outweigh the rights of others not to be forced to support it. If you can't support yourself, and nobody else is able and willing to support you, you die. Life isn't a fairy tale.
No child can support itself until it learns the means to do so - again your argument dovetails with that of Prof. Singer. Also implied by your statement is that the injured, sick, and elderly may also be legitimately abandoned to die, since they cannot support themselves. You are not the first to propose such a standard, but I am not sure you want to know where the modern precedent for such a point of view was established.