Using the same shaky logic that is used by the most patriarchy-promoting evolutionary psychologists doesn't make your guesses any more right than theirs, though!

The anthropological evidence that "patriarchy" is only 6-9k years old is weak at best. Social structures don't fossilize, nor do they imprint on artifacts particularly well. Closely related species include those with patriarchal structure (gorillas, chimpanzees) and those with a matriarchal structure (bonobos)...and those that are largely solitary or paired (orangutans, bonobos). Likewise, you can find a variety of reproductive patterns in modern-day and historical hunter-gatherer tribes, and in closely related species. You don't find high levels of social behavior coupled with monogamy among the great apes (gibbons are monogamous, but live in small groups), but you do in birds all the time, so it's not like the structure is fundamentally unsound.

Theoretically, under many social structures, sperm-competition is a poor way to choose a high-quality mate, except inasmuch as your offspring will have to engage in the same type of competition. This isn't to say that it hasn't been a significant part of human evolutionary history (much like other essentially exploitative strategies where there's a mismatch between benefit to the gene and benefit to the organism), but there's a very wide gulf between "it may have played an evolutionarily significant role" and "nothing else played an evolutionarily significant role".

Allele frequencies are sculpted over all kinds of timescales. (Here's one recent paper detecting substantial selection in enhancer regions of humans on all different timescales: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6686946/.) Changes aren't restricted to 200k years (that number is pretty arbitrary anyway--"modern" humans interbred with neanderthals occasionally, but neanderthals seem to have been distinguishable from the line that led most completely to modern humans for 400k years; mitochondrial Eve is very likely younger than 200k years, while Y-chromosome Adam is very likely older than 200k years, etc.). 6k years is plenty of time for big changes to happen, too. For instance, the Black Death seems to have had an impact (https://www.nature.com/articles/s41586-022-05349-x).

So I really don't see what the point is of emulating some of the least supportable styles of evolutionary psychology thinking, except if someone else tells an E.P. just-so story, you can respond with a different one and make the case that it's of comparable soundness.