You've acknowledged yourself that not all women have periods later in your post, so this 'simple way' is not effective as a test.
I think you are misunderstanding. Many women do not have periods, but the significance of not having a period is completely different for men and women.
If I don't have a period it is because I haven't yet reached puberty, I am pregnant, I am ill, I am taking medication that prevents periods, I am post menopause, or I have had a hysterectomy. Post puberty, if I don't have a period it's an indication that I will need to see a doctor, whether that is to get my contraception checked, to talk about HRT or to check that I'm not suffering from a health problem.
If a man went to the doctor to check why he wasn't having a period, there might be concern, but not about reproductive health...
None of this has anything to do with my identity.