In certain parts of the fandom, there's a real push back against the notion that Eowyn was left behind first and foremost because she was a woman. That Eowyn was denied the right to ride to battle, or was forced into a domestic role, because she was a woman. Or indeed, that her depression was directly caused by the choices other people made regarding her because she was a woman.
Fans will accept generally that countries like Rohan and Gondor were patriarchal (although they might avoid using the word "sexist"), and will acknowledge that gender roles were at play, but when presented with specific acts of sexism from characters they admire, like Theoden or Aragorn, they shy away from it, they try to find alternative explanations, they try to remove sexism from the narrative. Actually pin pointing moments of sexism from heroic characters is something they resist, even though they wouldn't necessarily deny that the characters exist in sexist cultures.
It seems that they are unwilling to fully acknowledge that sexist societies are sexist because of the choices and conduct of those living within the societies, that agents within those societies perpetuate sexism by making choices that reinforce it. They'd rather shrug the sexism off with a vague "it's just what it was like back then".
"It's just what it was like back then" comes up if you look critically at a character's sexist actions. We're told we can't "judge them through a modern lens", as though everyday sexism only causes harm in modern day, as though the book itself doesn't examine the role gender and gendered expectations have on women like Eowyn.
And of course, much of it comes down to wanting to defend Theoden or Aragorn or whatever character is coming under critique. They would rather look for alternative interpretations, focus entirely on the non-sexist reasoning for their decisions and pretend that gender never comes into it, point out the times characters treated Eowyn with something approaching respect or recognition (which should be enough to dispel accusations of sexism, even when it is nowhere equal to what a male peer would receive) or use the traditions of their culture to exculpate the characters of all responsibility for their actions.
The result of this is that Eowyn ends up being re-written as a misguided woman whose sense of oppression was all in her head, that she was misguided and selfish (tragically so, because we can accept she is Grima's victim, just no one else's), that she was "redeemed" at the end of the narrative by "embracing her feminine role", and that her conflict with gender and gendered expectations are for herself to resolve, with no alterations of concessions made by others.
This, even though Gandalf spells out to Eomer that sexism played a hugely significant part in Eowyn's ultimate despair, and that Eomer himself, after hearing Gandalf, accepted this and reconsidered their entire lives together.