Just to be very clear - I understood the comment about Paul to refer to the general male dominance of much (though not all) of Christianity.
I also suspect that "churching" although Biblical in the sense previously mentioned, harks back to older times. I also suspect that the very nice custom of setting aside a "special time" has been hi-jacked & twisted.
Of course the RC is a broad church, and those of us outside it will often only come across it at its best and at its worst, as those in the middle keep fairly quiet.
I well remember a tiny (now defunct) convent attached to a RC school in an industrial town where I worked in the 70s. If we got kids in there, the nuns would give them breakfast, wash them & sort out any uniform problems, get them to school on time & give them tea after. If there were any struggling families with even a whiff of RC about them, a good-hearted young priest would come (at the Health Visitor's or Social Worker's request) and baptise the children at home! I told this tale recently at a social gathering, and one lady (announcing herself as RC) told me I must have been mistaken and this could not have happened as the church wouldn't have allowed it! I hope that young priest has his place in the assured in the heaven I don't myself believe in!
No need to talk of its worst.