^^ A lot of people use "patriarchy" the way "kyriarchy" is used. But I think "kyriarchy" is more useful because "kyri" isn't a well-known prefix. A lot of people think that "the patriarchy" refers to some kind of conspiracy of "partriarchs" who scheme to keep women down. "Kyriarchy" is, at least for the present, free of such associations.
"Kyriarchy" (or "patriarchy" if used in the same way) recognizes the intersectionality of various kinds of privilege, such as racial and class privilege. Some people care about just one aspect of the kyriarchy, such as gay men who just want male privilege, or white women who just want white privilege. But opposing the kyriarchy means taking a stance against the overall system of oppression in society.