You seem like a generally thoughtful person, so I want to make sure that I respond as thoughtfully. I think feminists are right to look on the attempt to replace feminism with egalitarianism with a great deal of suspicion.

There are a number of different reasons for this.

First, because I think the history of egalitarianism is incredibly problematic in ways that feminists and proto-feminists have been engaging for the last 200+ years. It is pretty historically clear that the egalitarian movements that produced some of the most important documents on equality (such as the Declaration of Independence and the Declaration of the Rights of Man and the Citizen) were based on granting equality to a relatively small group of people while intentionally excluding large numbers of other people. The right to vote is considered pretty fundamental in the US, and has been since it's beginning, but women didn't have the right to vote, nor, obviously, did slaves.

Proto-feminists like Mary Wollstonecraft and the first feminists, who fought for the right to vote in the 19th and early 20th centuries, were explicitly criticizing the failures of the egalitarian movement for not actually being egalitarian. Feminism is now more prominent than egalitarianism because it actually had to fight for the equality that was falsely promised, and its history is divisive because of how entrenched the powers were/are that tried to prohibit and inhibit the equality of women.

So I think feminists now are justifiably worried that a new movement that purports to be about the equality of all people will in fact do more to promote the rights of people with more institutional power at the expense of people with less, because that is what the history of the egalitarian movement has been.

Another issue with the term egalitarianism is that it doesn't actually convey any information about where you actually stand on just about anything. Egalitarians in the 18th century were fighting against political systems that denied participation to everybody but the landed aristocracy who believed they had a divine right to rule. Very few people take those claims seriously now and it's a point that is basically accepted as given in the US. Everybody accept for the most unreconstructed bigots would probably identify them as egalitarians, so then the question becomes how do you actually achieve something? Does creating a more inclusive name actually make change more possible?

For me, I think the answer is no.

That's not to say that I don't think inclusivity is important. All movements should think about how to explain themselves in multiple ways for multiple audiences, and arguments that make appeals to inclusivity can be great. Obviously Martin Luther King is a great example of this, but he's also misused too. It's at this point trite to see some people using his lines about judging people "by the content of their character" as a rebuke to current incarnations of the civil rights movement while ignoring all the other parts of the speech and his advocacy where he explicitly praises the (nonviolent) militancy of the civil rights movement and his Letter from Birmingham jail where he rebukes "allies" who spend more time criticizing tactics than supporting their brethren's fight for equality.

The issue with replacing the term feminism/women's rights/minority rights/gay rights with the umbrella term egalitarianism is that I don't think it actually accomplishes anything to promote the cause of equality. In the US, I think 99.5% of people would describe them as in favor of equal rights for all. Yay equality! And yet I think inequalities clearly exist, and what's more I think inequalities exist that other people don't acknowledge are inequalities.

Take same-sex marriage, for instance. In 2001, 57% of Americans opposed same sex marriage and 35% supported it, and by 2015 those numbers had basically flipped. I think most people on both sides of the argument would say that they are strong believers in the importance of equal rights for all Americans, but what they disagree on is what those rights mean, and in fact many of the people who oppose same sex marriage use the language of equal rights (in reference to religion) to explain their support. The result is that you have some egalitarians who oppose same sex marriage and others who support it, and you end up with a divisive argument about what equal rights mean, which is what we already have. Making a cosmetic change to the name of the movement doesn't really accomplish anything.

And it's the same with feminism. Creating an egalitarian movement so broad that it would encompass every American wouldn't actually do much, because it doesn't change the fact that there are fundamental disagreements about what equal rights mean in practice. You'd still get the same arguments about pay, and some people would be convinced by reasonable arguments on either side of the issue, which is already the case.

Also, not every issue related to feminism is a matter of equal rights. Equal rights is absolutely the right rhetoric to use for things like voting rights, property rights, access to education, etc, but issues related to things like abortion or breastfeeding don't really fit into a category of equal rights and are more particularly women's rights.

The real issue for you seems to be that some feminists make bad arguments that inflame people and make them less willing to listen to the well-reasoned arguments made by other feminists. And to that, my response has to be, "well, yeah." That's how any large movement works. Lots of people are assholes, and convincing them to self-describe as egalitarians won't reduce the number of assholes working for the cause of women's rights. Some of the arguments will still be polarizing and will turn people away who might have been supportive. Sometimes reasonable people will also disagree about the value of being divisive. The right to vote was divisive at one point in time. Did that mean early feminists shouldn't have been divisive?

One point that I think I agree with you on is that there has been too much focus on what the word feminism means, gate-keeping around who is or is not allowed to call themselves feminists, and creating lists of shibboleths that must all be adhered to. The amount of mindless political posturing that occurs on social media is staggering. I would hasten to say that this is not something that is unique to the feminist movement or to progressive movements generally. It is a characteristic of every single movement that consists of more than one person. You can see it on the right with invocations of RINOs and cuckservatives. If you spend all your time trying to argue against the assholes in any movement, you're going to lose. The issue in the current generation is that social media amplifies the voices of the assholes, which sucks, but I think it's important to keep that sense of perspective whenever you see stories going around whose basic premise is, "look at how awful this insignificant stranger is for thinking [political opinion you hate]."

I think I've rambled myself about out of steam. If you've actually read the whole thing, thank you.