Well for one I don't think making any kind of statement against women's rights in the modern world via the medium of Lord of the Rings would even have been on Tolkien's radar - too often we find political hobbyhorses and latch them onto his work and they just don't prove to be there.
Eowyn's situation and how it came into being is much more interesting than some odious allegory of mysogyny as a matter of fact
How did she come to be there in the first place? She was originally conceived as a romantic foil for Aragorn before Arwen raised her head, and as such I suspect we'd have seen her in a very different role. She remained, however, and assumed a more complex part.
She's a member of a royal household and high status, and second in line to the throne. So the request that she stay behind to govern Rohan is one she might well have expected even if she were male, following the traditional 'heir and a spare' rule of thumb. Yes, it could have been that Eomer would return and take up his own place again, but it was also likely she would have been left a Queen in charge of a realm stripped of menfolk - a huge responsibility to bear. So I don't think that the request she stay behind was a cut and dried case of 'sexism'.
Her sword skills have evidently been gained previously as you just don't pick up a blade one day and instinctively know what to do with it, that only happens in films
It's not entirely unfeasible that Rohirrim women knew some basic skills in self defence and basic combat - they would have homesteads to defend after all when their men were on military service. And Tolkien isn't shy of throwing us a few powerful, feisty females when he wants to, so the argument he hated the thought of a 'shield maiden' on principle doesn't cut any ice with me.
But then my final thought about Eowyn is this reason why she decides to go haring off into battle. Ostensibly, it's because of a horrible case of unrequited love for Aragorn and her sense of desperation, but I always see more to it than that. She loves him as an incredibly idealised figure, admires the hope for military success he brings, is stirred by the promise more of
victory than of romance.
Eowyn in this view isn't even a woman at
all, she's the young Tommy, stirred by news reports and posters of Lord Kitchener into signing up and taking the King's Shilling. Signing up for the trenches despite knowing that death is almost inevitable. She's the boy warrior following the idealised captain over the top despite being ordered to stay behind. The kind of figure Tolkien, sadly, will have recognised all too well.