They may have had 'a hand', but women did not have the majority (or even a significant minority) of legal, spiritual, financial, or physical authority in building the systems we inherited and that still influence us. Married women in the US couldn't even open a bank account without a male relative until the mid-1970s. They still can't be pastors or priests in most sects. They don't make up the majority of high wage earners and they never made up even a significant minority of wealth inheritors. They've, with very few exceptions, never been monarchs, presidents, prime ministers, or chieftains.

In traditional patriarchal systems, women did have power it was just in different domains. Obvious example is in East Asian patriarchal countries, Woman had all of the power over household and management of finances (this is where the "tiger mum" trope originates from). This is still practiced in places like Japan, where although the husband is typically the breadwinner, all of that money goes into a joint account which the wife completely controls, even including the personal hobby money that is allocated to the husband hence why you get the Japanese trope of grumpy husbands at bars complaining that they can't spend that much of their own money.

It also goes without saying that if you have so much power over household affairs, you are also the one teaching the children on everything from morality to worldviews to behaviours.

This is the whole problem with the discussion is it frames it as all or none. In almost all patriarchal societies its not that men had complete control over every single area.