Sure, I completely agree with the first part, that's why I said removing religious voices from this debate would be very helpful.
The problem with the bolded part is that these organizations define what it means to be christian for huge swathes of the christian community, not you, and they don't agree with you. The Humanae Vitae, which is still in effect and surely applies to everyone who wants to properly follow that religion, is an example of that.
Agree again, but parents generally aren't writing policy papers which tell their followers that contraception is intrinsically immoral and evil, or that condoms don't help in preventing HIV for another example.
I don't mean to pick on one religion here, this is just the one I'm most familiar with, but in general preventing pregnancies is not in the interest of any religion that wants to increase its numbers. Perhaps someone familiar with other religions can tell us if it's the same for them.