I don't think most people living in other times & places privately agreed with their society's public morality, to the same extent that we do today.

For most of history (not prehistory), there was no option for public debate or even for openly stating opinions. Morality was normally handed down from above, from the rulers, as part of a religion. If those people had an opportunity to live in our society and be acclimatized to it, many of them may have preferred our morality. I don't believe the reverse is true, however.

This doesn't prove that our morality is objectively better - it's impossible to prove this, by definition - but it does dismiss the implication of the argument that "you like today's morality because you live today". Only the people who live today are likely to like their time's morality.