The idea here is that there's some condition implied or it's about a hypothetical case. "If X, she would not have become old."
For instance, if she had lived in the Middle Ages, she would not have become old.
She died as an infant, but she would never have grown old anyway because there was a huge famine a few years later when everybody died anyway. (sorry about the horrible example).
Maybe the problem is not with would but in the other part of the sentence, I didn't like the choice of verb now that I wrote them. In Swedish, bli gammal often means 'live until old age' but I'm not so sure that the English version can carry that implication.
If you say She should not have become old that more or less means the same as She ought not… The best translation of that into Swedish is borde. The sentence above means She would not have become old – some condition is implied. For instance, if she had lived in the Middle ages, she would not have become old.