Where do you think the leisure for civilization and all its benefits - including longevity for both men and women - came from? Not from a world where “women did not have a lot of choice when it came to a man wanting to initiate sexual activity.”
The hypothesis is that the “hidden” fertile period of human women was the force that encouraged the brighter man to stay close to home, to better ensure that her children are his. If this is true, then someone was having periods.
Leisure for civilization came from agriculture. When humans were free from having to track down food on a constant basis, and turned to the tending of crops, contemplative time for those unprecedented-on-earth brains was a byproduct.
One interesting thing you mention--a man wants to be sure his children are his. My hypothesis is that in the pre-agriculture period, there was no way of even having the concept of "my son". Since mating would have occurred within small tribal groups (most probably the dominant male would have sired most of the children anyway) there would not have been "resemblance" issues.
What changed this? Agriculture, and the need to domesticate large animals to till fields. The upper body strength of men made them the natural herders and keepers of these beasts, and fences were developed to keep them penned up. Surely some prehistoric herdsmen noticed that when a female animal was kept penned up without access to a male animal of the same kind, she was "barren". Introduce a male animal into the pen, and the barrenness disappears.
Surely, this had ramifications for human breeding, as well. In order for a man to have the concept of "my son", it meant that a particular man had to have exclusive sexual access to a particular female. At that point, the concepts of virginity and monogamy became instituted.
This only happened about 10,000 years ago, give or take a few thousand. It's a very short space in the span of human evolution. Besides, even then, a female would have been partnered off to a man about the time of her first menses, and then would have stayed pregnant or breastfeeding until her death, which would likely have occurred as a result of childbirth. She wouldn't have had as many menstrual periods as modern women, who reach menarche much earlier than our society considers them ready for marriage.
We can disagree about the reasons, but it's pretty clear that women today, on average, have many more menstrual periods than women did even a couple of centuries ago.