Part of the problem with religious belief exemptions is that the Catholics run a metric fuckton of hospitals and orphanages. The thing about Catholicism, as ... a Catholic (albeit one who is usually disowned by fellow Catholics), is that it's an absolute system. "Purity" as an idea or descriptor doesn't exist in many cultures! And Catholicism specializes in 1 thinking globally and 2 thinking in centuries. The little details don't matter as much as the big picture, and the power of the big picture cannot be changed (not in a way that's visible within a human lifetime, anyway).
Catholics do a ton of "civilizing" things in the world. One of the big problems with Catholicism is that it's basically "give you a sweet deal in exchange for your identity go brrr." The argument that the religious groups that are anti-abortion don't care about the person that come out later is false for Catholics. It's just that the way they care about you may not be a way that you particularly enjoy, and may require that you trade off other things you really value, such as your gender or sexuality or spiritual beliefs, in exchange for that care.
Also, when you say "please stop 'caring,' this is not good for me," the Catholic church says "no

" That's kind of their whole deal, for better or for worse! It's their idea of what God is like, so it's the style that they try to imitate.
It's important to contextualize the Catholic abortion position with the fact that the Catholic church is
also anti death penalty, which gets them in trouble with all the other Christians. It's a twisted viewpoint, but mostly a fairly consistent one. What they don't do with secular law, they do with religious law and an intense education in guilt. I grew up being told it was more important to be good than happy; my mom grew up feeling like any extra resource she had, from time to money to anything she enjoyed at all, had to be given away because there was always someone needier.
Catholic belief is always about looking for ways to give away more and more of yourself to the poor and a little extra for the church. The emotional argument that it's not a legal obligation to give away blood, organs, or whatever else is not an argument that's going to be especially effective here. The effective argument is for secularism, "you may believe that but we need THIS for the functioning of our government," which is the established, centuries-old solution. The church presses on the state and the state needs to push back. The church may not especially like this, but they do understand it. They want to have a positive relationship with other organs of authority, after all.
Oh, and if you're wondering about the whole tithing thing, give them a little money and they'll make you feel like you matter and like you're a vital piece of something bigger and better than your little old self. If you want secular society to be less vulnerable to the Catholic church, you need secular society to care at least that much about the least of these, which. Trust me, they don't.
When you have humanist chaplains ministering to people on every death row, you might be able to get rid of the influence of the Catholic church. But we're not there yet.
Now, something else to note is that this anti-abortion push, although we have 5 conservative Catholic justices,
actually is not spearheaded by the Catholics. It is an Evangelical Christian thing. Overwhelmingly. Check your statistics! That church is not against the death penalty, they believe in hell and that God is all about fire and brimstone JUSTICE and that bad things don't happen to good people. Good people are rewarded in heaven and on earth. If you are in a place to need an abortion, it's because you need to be punished, and punishing you is more important than anything else, such as what happens to the child after it comes out.
So: I don't know why people can't seem to place the blame where the blame should be, but the Satanist churches have the right idea. This is a religious argument about the role of bodily autonomy. If we wanna win, we need to counter the Evangelical argument and change the way that people think about the body and society.