Even in the word itself "women" is treated as a special modifier of "men." Men + wo = women, fe + male = female, etc. Men are considered the 'default human." This is one of those language evolution things. "Man" was originally gender neutral in old English, meaning person, and the genders were wif(man) for female and wer(man) for male.
That talking point presumes it's all men and only men who are against abortion. Plenty of women are as well. We live in a society where everyone can vote. If women all voted together, they could probably get whatever they want.
I don't think it's "Puritans" who have the double standard here. It's liberal/leftist feminists, because they see things in terms of an oppressor vs oppressed dynamic. Men sexualizing women is bad because men are the oppressors. Women sexualizing themselves is good because they are the oppressed. Sexuality itself almost doesn't matter in the equation, it's about power.