Isn’t this question somehow? Feminism was never initiated in lieu of gender equality. Gender equality is proposed as a more consensual, neutral term supposedly meaning the same thing. Feminists hold on to the word because of its historical value, but also because they believe that the current social organisation in which we live is still treating women, specifically, badly. As such, it is important to keep the name “feminist”, not because they don’t care about men’s rights, but because achieving sex equality means, to them, freeing women from the violence and erasure that they still endure. For feminists, men can be structurally screwed, but not because they have a penis; what the current gender beliefs our societies hold doesn’t try to turn them into victims, but encourages them to become entitled and violent (which still sucks, still dehumanises those who yield to that pressure, but that means it is not their lives that are directly threatened by interactions with the other sex.)
That’s precisely what disturbs people who don’t like the word feminism. They don’t agree that we live in a world where women tend to be structurally screwed specifically because they were born with a vagina. All of this is open to discussion, of course, and much has been written already to describe or reject the notion of a patriarchy oppressing women.