I don't like agreeing to disagree; I think there's merit in impassioned disagreement - perhaps we may never change each other's minds on this, but there are others who will read this back and forth and who may move one way or the other. Two people respectfully arguing their cause is what democracy is about - and for what it's worth I think you've fought your cause admirably.
The reason I'm erasing the meaning of the word today is not so it loses its meaning, but so that it becomes a pragmatic tool. Feminism has achieved a lot of good, it has done some great things - but it has also overstepped in the wake of the #MeToo movement into something occasionally approaching misandry - as I outlined in my Amber Heard article. That doesn't diminish what it has accomplished BUT, it does make progress from this point much harder. When combined with an emergence of something I call 'femagoguery' - which is women writing for other women about how toxic and/or awful men are (and is rife on Medium), you can see why things might not move much further.
Gender egalitarianism takes a holistic look at a problem and offers solutions to both parties by framing it objectively. As an example, let's talk about the societal issue of rape. It disproportionately affects women (though men are also raped) and the conviction rate is far too low and the process is far too slow and far too many rapists walk free. There are also women who falsely accuse men.... the immediate feminist response is 'that is disproportionately small and not an issue!!!!!' - which is a gendered response based on being female, because although there may only be a disproportionately small number of houses that catch fire, we still have fire engines.
How do you solve that? Well... I'd argue that you need individual specialised courts, with individual specialised lawyers and judges - people who deal only with cases of rape and/or sexual assault and you have a mandated minimum time frame in which the case should be heard. I'd also suggest that every city has a specialist service dedicated to providing an emergency service to any woman (or man) who has been raped and this is highly publicised service capable of collecting DNA evidence to be used in court.
So far so good.
I'd also suggest that these courts hold an infinite gagging order on both parties until the verdict. This means if the name of either party is revealed in the press or on social media - then those people will be held in contempt of court and may go to prison. If the accused is found guilty then he/she may have their details splashed in the press. If it turns out that the accuser has fabricated the story, then he/she can also be named in the press. Other than this, the whole court operates quietly, quickly and efficiently. It also provides outreach education, training and provides a feedback loop into parliamentary procedure for quickly updating the law when necessary - eg, revenge porn, cyberflashing etc.
Why am I framing it this way.... because women get bogged down with wanting to catch rapists (and dismiss men's views as unimportant) whilst men consider women to be acting in bad-faith (because there are highly publicised cases where this is the case). Everyone thinks about the issue based on their framing, instead of thinking about a global solution which works for everyone.
In this specific case, gender doesn't help. The outcome I have outlined is the reduction of rape in the long term... via the catching and prosecution of more rapists. Women's suffering at the hands of rapists far exceeds that of men, but framing it as a female issue and highlighting female suffering here hasn't engendered enough social change. These changes would disproportionately help women in society, but they also address concerns that men have - with more to consider from both sexes. Instead of a tug of war, we need to agree some basic fundamentals and move forward accordingly.
That I'd argue is gender egalitarianism. Legal and social fairness for both sexes.