"Imagine a society where no one gets married, and everyone stays single all the time, constantly competing for partners, and most men are depressed and fatalistic because a few powerful men monopolize all the women. How much gets done?"
A lot as well. Because one of the ways men compete is by making money, which, as long as violence is criminalized, is mostly acquired through work and productivity.
"Remember that women don't care what men say, because they've evolved not to trust words. It's easy to lie with words, and women are very intelligent on most things lie-related."
I don't know about that. The proportion of gullible/credulous people in the male and female populations seems to be roughly the same. It isn't atypical to hear a woman go, "But he said he'd do this, he told me he'd stop doing that, etc." and then after a while give you something along the lines of, "I can't believe I was so stupid to believe it."