One connection is how to train your baby books I mentioned. The foundation most popular book is named "To Train Up a Child". They are literal Christian child raising books that teach how to property spank. These and similar books/approaches are recommended inside fundamentalist circles, unknow or controversial outside them. Christians openly advocating for these approaches is just normal in more fundamental circles (by which I mean I personally knew such people and they are not even all that much fundamentalist, just tilted that way).

And atheists and agnostics would not ever strike their own children, or their students?

They dont claim it is biblical requirement when they do that. They dont have organized promotion of those practices the way fundamentalists do.

Also "I am going to homeschool because my kids might have meet christian in public school" is not really a movement among atheists.

you don't recognize there's any sort of historical pattern between religion, at least w the more extremist parts, and punitive punishments? conversion therapy, Indian boarding schools, child wilderness camps, Catholic sex abuse scandals are just a few that pop to mind