A man can have a policy of, say, opening doors for women and still be a selfish lazy jerk, expecting his wife (real or hypothetical) to do all the housework, all the child-rearing, all the emotional labor of managing all the relationships in the household, obtain almost half the income (but never more than he does because ego), etc. etc.

Chivalry doesn't prevent any of this. It ought to, ideally, indicate that someone is not inclined to do this, but nothing about it prevents this. It's not a binding contract.

Indeed, because humans are so good at accepting a whole package of behaviors at once, if chivalry gets associated with those sorts of behaviors, then chivalry will signal exactly the opposite of what it would if it were an honest signal.

But at a baseline, displays of willingness to equal out unfair burdens are reasonable to treat as a desirable signal.