What is your definition of rule, which you insist is unequivocally intrinsic to the meaning of the arch/archy part of patriarchy?
Okay, the definition I'm using is the ability of men to control women, backed up by a larger authority, such as the church, laws etc. So individual males may control individual women, but in the UK that's not sanctioned by a bigger authority, so it remains at an individual level (though of course having an impact on wider society.) A random man can try to tell me to cover my hair but he can't make me except by force. And in theory at least, if he tried this, he would be the one to get in trouble, because he's the one breaking the law.
I would agree that is patriarchy in action, but I don’t think it requires a government which is explicitly patriarchal for that to be the case. The sexual harassment in the street and DV in the homes in the UK is evidence that even the most pathetic man can rule over women, both his 'own' women and random women in the street, here in the UK, even though it’s illegal, and that the words patriarchy and patriarchal are relevant in feminist discussions at home.
I see where you're coming from but I think the legal status/authority is a vital distinction. It's not okay for men to harrass me in the street. Society as a whole doesn't sanction it. The media doesn't excuse it (as much as it used to). The law is against it. Now I completely agree that there are issues to do with sexual harassment and assault which probably have roots in sexism and misogyny. But it's not the case that any man can rule over me - in fact, in principle, no man can rule over me simply by virtue of being a man.
I think this might be a case of glass half full/half empty - what do you think? Because I feel very positive about the position of women in the UK generally, while acknowledging there are some big issues still to work on. I feel grateful to the women who went before me and secured equal rights for me. I value my freedoms and exercise them as fully as I'm able at any given opportunity. And when I see a threat to these rights and freedoms (from gender ideology) I feel determined not to have them taken from me.
Whereas perhaps(?) you feel that women are still in a parlous condition, despite having achieved legal equality. We are still influenced by the threat of rape, which often goes unpunished. And there are many situations where women struggle and suffer more than their male counterparts. I don't disagree with these things, but I think it's a bit one-dimensional. I think the explanation for these things being patriarchy/patriarchal systems is inadequate. Some of it is to do with sexism and misogyny and patriarchal phenomena, definitely. But it's only a part of the story. So the use of the word patriarchal to describe the situation, rather than being a useful analytical tool, seems to obviate the need to truly understand what's going on. E.g. it's obviously not the case that the lack of rape convictions is solely due to misogyny. There are other factors, e.g. the nature of the crime not usually having witnesses, which makes reasonable doubt a bigger influence; the role of alcohol in many rapes; the fact that rapes often occur within established relationships etc etc. It's complicated - saying it's 'patriarchal' doesn't help us understand the various factors. Or it doesn't help me!