Abortion, competing rights, and the problem with Thomson's violinist argument

I think the main problem with how people (on all sides of the debate) view abortion is that they often fail to recognize that there are at least two people involved. Many pro-choicers argue that unborn children should have no personhood status, while many pro-lifers are so focused on the rights of the unborn child that they are unable to make a decent counterargument against the whole "my body, my choice" thing.
Because of my libertarian leanings, I would be both pro-life and pro-choice, if those terms were used to describe issues other than abortion. I can totally understand why someone would be uncomfortable with anti-abortion legislation.
Thomson's violinist argument actually brings up some important points, and then there are similar arguments about bodily autonomy and self-defense. There are things that make abortion unique. First of all, abortion does not simply terminate an unwanted pregnancy; it takes away a life in a very brutal way. I know, I know. You're probably thinking "what about self-defense?" In cases in which it would be morally acceptable to defend oneself, at least one of two things tend to happen: either the person is defending themselves because their physical well-being is at risk, and/or the one that may be killed is intentionally trying to harm the person in a way that will scar them psychologically (i.e. sexual assault). In the case of abortion, the unborn child is innocent. Their intent is not to violate the pregnant person's bodily autonomy. Come on, pro-choicers are always pointing out that fetuses don't have (deep) thoughts or emotions (which is no excuse to kill a human being, or else we'd be killing sociopaths and those with intellectual disabilities). In most cases, the mother's life isn't in danger, either.
Although people can choose not to have children, it is important to consider that, when a parent has custody over a child, they are obligated to take care of them. In most cases, parents can give their children up for adoption if they cannot care for them. In the case of pregnancy, the pregnant person is the only one who can keep the child alive. This isn't about forcing pregnancy or parenthood or whatever; it's about the well-being of an already-existing unborn child. If a parent has custody over a child, they have no right to neglect or abandon them. They certainly have no right to abuse them, which is what abortion is. People have the right to abstain from sexual activity or use contraception in order to prevent pregnancy. They do not have the right to harm or neglect a child. It's fine to not have children, but it is wrong to abuse or neglect a child who actually exists.
When I say abortion is a competing rights issue, I am not implying that the unborn child "wins." I am saying that, while bodily autonomy is a right, it does not justify aggression.