In older patriarchal cultures, men went with men in terms of leisure activities and even sometimes for sexuality. Wives were valued only as bearers of children. If they couldn't do that, a certain group might be bred just to provide men with pleasure but other than that those men would see no need for women.
Just look at recent statistics--women make 20 percent less than men for the same position. Despite this being the 21st Century and not the Dark Ages, women are still discriminated against as pieces of meat to be used for sex and childbearing. If we truly respected women, they'd be treated like goddesses for their ability to carry on the species with childbearing. Without their ability to carry babies to term, the phrase "women and children first" has no meaning--men would trample women first for sport before killing each other in battle.
LOL, what the hell. Rome was not a patriarchal society at all. Patriarchal societies were very much against homosexuality. They didn't believe in it. Patriarchal societies are pretty much based on one thing -- what makes men men. Manliness is a key trait of patriarchy. If you were seen as gay, you were out of the club.
I do not believe women would become extinct if they didn't bear children, in a patriarchal society. I say this because women would still be found useful for household chores and otherwise taking care of men or supporting their needs. Assuming that women do not produce children doesn't eliminate their usefulness for sex either.
In a patriarchal society, women would not become extinct if they could not have children. Women are extremely capable in caring for themselves. They should be regarded on a similar level to men in all cases. This is why women would not become extinct in a patriarchal society if they could not have children.
If a society is patriarchal, it simply means that the society or government in its whole, is governed by men. It doesn't dictate that women somehow don't exist or don't play an important role in society, they are just not in charge. If there are no women, there are no men. While there ability to bear children imparts a lot of their duties in a patriarchal society, they also provide other things.