Why has this become a discussion on hitting? Chinuch involves a lot more than hitting your kids.
A parent is obligated to be michanech their child, even in public.
Speaking of chinuch in public…..
A woman once apologized to me for “ruining the chinuch I was giving my children” (her words). I was waiting at a red light and explaining to my kids that we should wait until the light turns green. This woman saw no cars coming and crossed on the red. My kids said nothing about it to me (although I was thinking of an answer). When we got to the other side on the green, the woman waited for us an apologized.