Vote, receive government loans, receive government benefits, work government jobs... you'd actually be surprised just how much of the everyday stuff Americans are supposed to receive from our benevolent federal overlords that is locked behind registering for the draft for men.
The amendment would also make it a lot harder to just brush off cases of sexual discrimination and harassment by women towards men, and would invalidate a lot of laws that make it impossible for a woman to be guilty of rape or that cause the cops to automatically arrest the larger individual on a domestic violence call, and a ton of other legislation and laws that objectively privilege women, which is another reason feminists don't really want that amendment to pass. But they do get a lot of mileage out of complaining that it doesn't exist... much as they continue to bitch about the imaginary pay gap and say we need a law against it despite said law already existing for decades.
To tie it back into TDS, it's really quite analogous to the whole impeachment grandstanding. They don't actually want to go through with the process, which would be an absolute negative for them on all levels, but they sure do want to talk a lot about it and use the talking to work the gullible into a lather.