Actually, it isn’t ingrained in many American cultural groups. I grew up in the old South where men only extended their hands first to hookers. Most of my families close non-Roma friends were orthodox Jews or Latino, none of them ever pawed at women, well hookers in bars maybe, but not women they respected. In fact, the only American cultural group that I know of where sticking out a hand at a woman is accepptable is the lower working class sort of redneck culture where the people don’t know how to act like polite members of society.