I may not agree with some, or even most of what the "sexism in games" movement has become, but on a broader level, it's probably a good thing it exists insofar as it's making people think about this stuff more. That support is contingent on the movement becoming dramatically more mature, self-critical, and careful about throwing around judgements and calls to action before the facts of a situation are known, and as you might point out, that's a big "if".
All of that said, there's one thing about the movement that I can't tolerate: the tendency to broadly attribute thoughts, motives, and preferences to men. It's unfair, fallacious in countless ways, and basically forces men into a corner before a discussion has even begun. I can't claim to speak for a broad swaths of people, but I imagine a some (certainly not all!) of the criticism, anger, and general hostility toward the Anita Sarkeesians of the world (and those who support them) is derived from a reflexive dislike of having motives and labels (many of which basically imply most/all men are neanderthal idiots) attributed to oneself.
It’s become standard practice to claim that almost anyone who questions the unproven idea that contemporary gaming is inherently sexist (or really, even anyone who pokes holes in feminist gamer arguments) is a misogynist. Misogyny has become a label that people feel comfortable using in the same way one might use "idiot" or "asshole", but it's a far more specific and vilifying word, because it quite literally means that one hates (or at least strongly dislikes) women. Pointing out that many of Anita Sarkeesian's arguments are based on false, pre-established, inherently skewed premises does not mean that I hate women. Pointing out that forming angry internet mobs accusing people of "misogyny" based on something you read on Twitter is irresponsible and borderline libelous does not mean I hate women. Pointing out that this movement desperately needs to call out some of the laughably illogical and fallacious arguments its de facto spokespeople regularly trot out does not mean I hate women. Nor, for that matter, does any of that mean I'm sexist, which like misogyny has a meaning that many who use it are oblivious to.
In fact, for many of us, it's because of our belief that this is an important issue that we're so hostile toward those who are making it look juvenile, navel-gazing, and incapable of self-criticism because of the spectre of being thrown in the "misogynist" discard pile. It's not your right to tell us that we disagree with you because we hate women, just like it's not my right to tell Christians they hate women because their churches oppose abortion or women in the priesthood -- and really, that's a way more fair criticism because it's based on verifiable evidence and not blind accusations or circular reasoning.
Moreover, don't tell me I like certain games because they're a "male power fantasy", because they reinforce the idea of women as "property" to be fought over, or even because I like the idea of playing a buff, "masculine" hero (hint: I don't). A lot of these criticisms imply that men are incapable of self-reflection, are driven primarily by neanderthal, sex-fueled desires, and are essentially incapable of empathizing with and respecting the opposite sex. They imply that a plot about a man rescuing a women is effective because the men playing these games are neanderthals, not because the people (not just men!) playing these games have a human, non-psychopathic desire to help those in need -- especially when, as is so often the case, said "damsels in distress" often have broader importance to the fates of large groups of people, and often display admirable strength (both of character and body).
In so many cases, when given the choice between a reasonable neutral/positive interpretation and a grasping negative interpretation of a plot, character, or mechanic, the "sexism in games" crusaders insist on the interpretation that fits their preconceived notions of male psychology and allows them to blindly label men as psychopaths or women-haters. It's ironic, because the exact same feminists who insist on broadly labelling men are extremely (and usually fairly) opposed to labelling women. I'm not going to start listing examples stereotypical female labels because it pains me to write nonsense, but I'm sure you can fill them in yourself.
Who gave you the right to define my enjoyment and interpretations of games? Who gave you the right to reinforce the tiresome media trope of treating men like bumbling, oblivious neanderthals in the same breath with which you decry tropes that you think demean women? Who gave you the right to tell me I hate women? Who gave you the right to essentially accuse me of psychopathy? In short, who gave you the right to tell me what I think, and why I think it?
By all means, continue to criticise games based on whatever nebulous standards you want -- if you're careful about it (not holding my breath), feel free to criticise those who make them -- but stop claiming to know how people relate to them. It's a presumptuous, unfair, juvenile, hypocritical, and just plain shitty practice that needs to stop.
P.S. No, I'm not trying to write away any of the legitimately sexist, awful, and yes, occasionally legitimately misogynist stuff that people (not "gamers", but people) have absolutely been responsible for. If someone says something awful, you've got every right to hold that against them, but you don't have the right to hold that against every male who plays video games. My issue is not with pointed criticism, it's with the reinforcement of broad, unsupported stereotypes and a culture of shaming people for thought crimes.