Menstruating women don't lose nearly that much blood.
Even if they did, the hormonal effects would likely swamp anything else. Which is a huge problem: women are routinely excluded from studies to avoid that, meaning we have no idea what the effects are on women.
This has been done. Women seem to have certain health benefits that stop after menopause. Reading about it was the first time I wondered whether blood letting made sense.
God, this is so ignorant, the hormonal changes (loss of estrogen) are the cause of increased risks for heart disease and osteoporosis and changes in metabolism post-menopause. Nothing to do with not physically losing blood, FFS.
There are likely multiple causal factors behind the health differences. Hormonal changes are one piece of the puzzle but so far no one has conclusively proved that physically losing blood has zero effect. The research just hasn't been done yet so we can't definitively say one way or the other.
There is a significant difference in the rate of major adverse cardiac events between menstruating women versus men and post-menopausal women, even after controlling for age and other factors. The periodic blood loss might account for at least part of the difference although the exact mechanism of action hasn't been clearly established. So it's possible that donating blood (or bloodletting in general) could have a preventive effect.
I looked up the amount of blood lost due to a menstruation cycle, and the answer is around 50 ml.
OP's linked paper has "the iron-reduction patients had 300ml of blood removed at the start of the trial and between 250 and 500ml removed four weeks later."
A blood donation removes 500 ml, so about a year of menstruation all at once. You can donate every two months, besides.
So, yes, if there is an effect then we might expect the magnitude of the effect to differ. Or else we'd expect a paper cut to also have the same effect.
Even if they did, the hormonal effects would likely swamp anything else. Which is a huge problem: women are routinely excluded from studies to avoid that, meaning we have no idea what the effects are on women.