The 'giving women control over their own bodies' argument in favour of allowing abortion is not the best or most reliable argument, imo, of itself and on its own. It can be part of a more sophisticated case, yes. For example, it could be (and often is) that women are given control over what happens to the fetus in their bodies, up to a certain stage of development, barring exceptional circumstances (when that time limit may be exceeded). Which imo they obviously should be (given that control).
The question of when the developing entity changes from being part of the woman to something distinctly not part of her (even if still inside her) is not easy to answer. In one way, it's 'part of her' until it is born, but it could equally be said that it is merely 'in her'. Another option is that she and the fetus are a shared system.
No.
I understand that you (and some other people, often male people) reject the argument that a woman should have control over her own body. Her own personal needs and wants should be made subordinate to the needs of an innocent developing fetus (most morally acceptable) or, more obscurely, to the needs and wants of a man, a family, or society.
This, in many respects, is not different than the notion that at least some women must be made available, willing or not, to satisfy the needs of (some) men for sexual gratification. Or to cook, clean, bring coffee, sort mail, provide ‘menial and therefore poorly compensated labor for men who are made for better things.
Any person deserves to have autonomy and control over their own body. No one should be compelled to do so much as donate blood, much less cede their entire body and health and education and career and personal development to serve the needs of any other being unless it is their express desire to do so.
I am not familiar with the abortion laws in other countries but in the US, it was established that women have the right to terminate a pregnancy up until the point of viability of the fetus, after which that right MAY be restricted to certain circumstances such as the life and health of the woman.