I don't think it's as simple as you portray it. It's not a binary issue of when a fetus becomes a child, and I don't believe that's where the argument is centered around. The pro-choice argument is a argument of women's rights, while the pro-life argument is an argument of when a fetus becomes a "life" and the rights surrounding that "life".
The pro-choice side generally doesn't care too much about when this fetus becomes a "life", nor is it really a cornerstone of the stance. I don't care when people want to call the lump of cells a "life", if the woman providing her body, her nutrients, and her life for it decides she no longer wishes to do so, that is her choice. And I believe most pro-choice people echo that sentiment.
The reason the sides argue for different reasons is because the pro-life side knows their argument cannot be centered on the idea of anti-women. It would gain no traction. The same reason exists for the pro-choice side. If they argued when the clump of cells becomes a "life", it would be hard to argue against pro-lifers because its more of a philosophical/ethics questions.
So I disagree that the argument is centered around when a fetus is a life. I believe one side focuses on women's rights and the other side focuses on when a fetus becomes a life and the rights surrounding that fetus.