First of all, it's not an exclusionary thing. Most actual human beings I know who identify as egalitarians also identify as feminists.
It's really all about analysis.
Throughout history, there have been plenty of people who call themselves egalitarians who have no problem supporting slavery, or the legal oppression of women because they lacked any kind of analyses that these things were wrong or perhaps denied that they were unequal (slaves have it pretty good, right? Sure, women are equal because they raise kids and leave politics to men, right?).
I have nothing against egalitarians. It's a fine enough description, though egalitarians aren't exactly known for pushing to change things and create equality as a group.
More often, on the internet, anyway, it's a description used by people who want to be seen as "neutral." Maybe that's just my experience, though. So egalitarianism doesn't really "fight for equality for everyone" so much as it just sits there.
Feminism includes an analysis that our society has tremendous gender inequality problems that hurt both men and women - keeping women mostly shut out of the most influential political and economic positions and creating gender roles that lock men into particular roles as well.
Intersectional feminism fights to eliminate sexism, but also notes that there is a complex landscape of other people also combating inequality related to race, class, sexual identity, and more. We can fight sexism while also being a good ally for example, to the Movement for Black Lives. And that's a much more thoughtful way to fight sexism than a narrow focus on just white, straight, middle class Feminists. We can push ourselves to be more thoughtful about the struggles of others even as we're fighting our own.
Does that make sense?
I appreciate your curiosity.