Corporations have enough personhood rights to have a major influence, if not the largest, influence on voting in America. In addition, their personhood can live "forever" without the natural death turnover actual people go through providing demographic refreshment.
They don't carry the same responsibility as actual people, however. Also, their ability to influence politics in even the broadest of senses is a major flaw in our country's design. If you're trying to run a republic, it does not make sense to move the incentives of the governor away from the governed.
So a woman has the right to choose whether or not to carry a pregnancy to term? "Rights" is a pretty meaningless answer without elaboration. Is a child conceived in the US a citizen then from the moment of conception? Or are the rights you speak about not the rights of USians?
Rights is a blanket term, you still have to define what is and is not a right. People have a right to life/liberty/property/etc, but not to the extent that it allows them to directly deprive others of those things.
While one might not conclude it perfect, we do define rights for people though. That's not the issue at hand. The issue at hand is when a person starts being a person. That's something we need to decide and put into law, probably at the state level based on the construction of the US constitution, if it hasn't been done yet.
There's no point in getting into the weeds about privacy in medical procedures, and there's no point in getting into the weed about specific rights. Everyone here would broadly agree that people, including young children, have a right not to be murdered (aka right to life). The only meaningful question is whether the life being destroyed is that of a person. Decades ago, the courts punted on that determination, and you could make a strong case that this is not for the courts to decide. But it is THE question that matters to this discussion. Without answering, you cannot reach the starting line for determining abortion policy, and you cannot make a coherent argument on the morality of abortion (either preventing or allowing them). You NEED to define this for the question of abortion policy to be functional. And by you, I mean individual voters in each state.
Given its causal inferences, it is also noticeably different from gun control. Gun control is increasingly about punishing/criminalizing possession, when it is generally not possession we should care about, but crimes committed. Depriving non-criminals of rights/property in the name of reducing crime has obvious pitfalls wrt constitution, but also some less obvious ones (inconsistent standard to other laws, or worse, using this new standard to shackle people more broadly).