Yes. It prohibits the disenfranchisement of any citizen based on sex. At the time that really only had impact on women, but the amendment doesn't specifically grant voting rights to women.

The post is clearly a joke because they start by appearing to advocate for an awful thing, and then turning it around at the last second by advocating for the awful thing for the people who most often actually advocate for the first bit.

It's not super funny or poignant, but it's obviously a joke.

Agreed, men could spend more time at the gym if they didn't have to consider any sort of political implications. It would be better for them.