That argument could be made for any form of education... Especially ones that lean towards one political party or another. Why have schools at all if the kids have to choose what schooling type they get from experience or knowledge they don't yet have? Don't take away one valid schooling method for an illogical reason. (it seems like a logical argument, but take a look past the surface and you will see that it is not)
As it is, every schooling system currently in existence is a form of indoctrination. Which school you go to determines what type of indoctrination you receive. Typical public schools teach many theories as fact, (like evolution) and generally push the students towards the Democratic side of the political spectrum. Typical Christian schools teach the bible as fact, (there really isn't any scientific counter-evidence that isn't itself highly controversial) some theories as fact, (like creationism) and generally push students towards the Republican-Independant side of the political spectrum. Why do you think it's a good idea to ban any of it?
It is an interesting question—what is and what isn't indoctrination? I can't really speak for American schools, but I can't say the schools I've been to have really pushed me towards a certain political ideology. They have of course taught things that I disagree with, but those are few and far between. I think the main issue I have with religious schools is that they try to teach something that should be taken as moral advice as the truth. We've done religious studies in the past, from around grade 3-5, and that mainly focussed on Christianity. They didn't, however, teach it as fact. Most if not all my friends are atheists or agnostic, and I think that comes from the way religion is taught in schools over here. Is one religion better than another? It's all relative to what you're immersed in while growing up. I can say subjectively that I disagree with certain religions like Islam for their teachings, but I can also say the same for most religions. Do you think we should have more Islamic schools? Or do you think Catholicism should be the only religion taught? I see religion more like moral teachings, but I think people should be free to make their own minds up about those issues instead of being taught them in school. Having mandatory religious classes would be a smart move, or just teaching more about them in classes like Humanities. That way people can make their own decisions on what's right and what's wrong.
But to play devil's advocate, what's the difference between teaching morals in schools and teaching a certain religion as fact? If I say to someone "Killing is wrong" do I have to back that up with facts right there and then? Or are certain issues 'obvious' in a sense? I think some topics may be, but of course that could just be side effect of the society I've grown up in. Should every opinion have to be backed up with the scientific method? I think that's one of the other issues I have with how certain religions are taught in religious schools, they say things that fly in the face of logic. How can someone say that the Earth is 6000 years old when we have proof of it being older? Of course you could always use the rebuttal "God just planted those dinosaur bones there when he made the Earth", and there's no real way of arguing with that.
On a more personal note, do you think all religions are equal? Or do you think Catholicism is objectively 'correct'? Is there something specific that makes you stay Catholic? I don't think I've ever had a belief so strong that it lasts my entire life, I tend questions issues too much.