Why can't there be a way to remove a living fetus from a woman without killing it?

Every woman that gets abortion does so because they don't want to mother a child they didn't plan for, abortion though involves killing the human fetus. Out of sentimental reasoning, imagine if the state took tax payer money and funded researchers to come up with a surgical procedure that can remove a living fetus from the womb without killing it, putting said fetus into an appropriately sized tube or container of some kind of liquid with warm water and given sustenance until it becomes a fully grown newborn baby. If this became possible I'd like to know why this would fail? If you want the fetus out and not be a mother why would it matter if the fetus is alive?
Why can't there be a way to remove a living fetus from a woman without killing it?
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