Now with all of that said: Why is it such a problem with Feminism?
The first answer is that it is Easy to Ignore.
Most women in Western Society grow up surrounded by the Little Things. It is the background noise of their life. As simple and common as the high pitched ringing you hear in your ears right now... now that I have you specifically listening for it. Whether you're listening to it or not, that ringing is always there, just at the top of your register. Sometimes if you focus real hard you can hear other sounds as well. Just as steady. Just as clearly. But only when you recognize they're there and pay attention to them.
Now with that background noise comes something far worse: Internalization.https://www.youtube.com/watch?v=XP3cyRRAfX0 Linked for child
I'm sure plenty of us have seen this commercial. And, together, the statements of the mother and father tell a very discrete story of discouragement. But each, on it's own, seems perfectly reasonable. A parent simply being protective of their child, her clothing, or their own sanity when a school project gets "Out of hand". But none of those interactions exist without the others because the girl experiences all of them. And she has since before she had an understanding of words. Or, rather, the girls that the character represents that exist in our society are bombarded with these messages.
By the end of the video we're given a rather ham-handed idea of how the girl views herself.
The ham-handed ending, however, is a fairly good example of what internalization of sexism looks like. It's very likely that character will grow up and impart the same values on her children, probably just as unintentionally as her own parents did.
And what's worse: she'll defend each lesson and each value as perfectly valid because in the moment itself it seems like a fine thing to say, but the impact will be cumulative. At that point she's gained enough tolerance for the abuse that it no longer registers as abuse that she's placing upon her daughter to ever so slowly crush her interest in science and nature and anything otherwise coded for "Boy" in our society.
The second answer is: It's Easy to Dismiss.
It's not a big deal. It's irrelevant. It's tiny and pathetic and small and you shouldn't bother with it. You'll hear that a lot when you talk about any sort of Social Justice, and feminism is no different. But while you'll hear that about stuff that is considered a bigger deal (like the rampant sexism in gamer culture) the protestations get louder the more specific an example.
Because it is so easy to separate that example from all the others. To say that "This" one is too small to care about. That there are bigger matters to handle. To deride an argument as being miniscule and not having a bigger target.
What these arguments tend to miss, though, is that small, casual, common, thoughtless forms of sexism are far more pervasive than overt examples... but they are no less important specifically because they are so very common. They form an unconscious and simple form of dehumanization and minimization of women that is so ingrained that even the thought of examining it is viewed as a perverse waste of time.
Which brings us to the third reason: Self Inspection is Hard.
Looking at oneself honestly, and judging the motivations behind actions that are performed on a routine, is difficult. Seriously criticizing those motivations under the question of sexism (or any other pernicious form of bias for that matter) is even harder. Because we do not think of ourselves as sexists. And to self-criticize on the basis of sexism results in two common ends: Either recognizing that some of our social cues that we've internalized are sexist, or vehemently opposing even the idea that we might be sexist and rejecting the entire experience out of hand.
On some rare occasions people can, presumably, honestly and sincerely look into themselves and find no internalized bias, but I've never seen it happen or heard of it happening.
I have, honestly and sincerely, looked into myself and found bias I didn't recognize was there. I openly admit it. And it is only through honest appraisal that we can work towards bettering ourselves, or at least shielding others from our internalized biases.