Vivian's a "real gamer girl", which implies that there's a fake kind.
One can fake a passion for anything - it isn't gendered. Anyone can pretend to be passionate about video-games, that is what a "fake" gamer boy/girl/x would be. It isn't based on sexism - it's based on the notion that people in anti-gg don't actually play games, and are simply trying to pedal a narrative with no care for actual gamers. In my view this was reinforced by the whole 'gamers are dead' thing, which really set the notion that some journalists/critics aren't gamers into motion.
As for your bullet points:
Games are fundamentally toys. They CAN and do have messages or stories behind them. But fundamentally they are about enjoyment by and large, with niches that are about both enjoyment and the message [I.E bioshock infinite - philosophy ect themes] or just the message [these tend to appeal to a much smaller audience.]
Sexism is a very nebulous term. Is a female avatar inappropriately dressed sexism? Is it over-sexualisation? Is [over] sexualisation sexist or not? Is hundreds of disposable male avatars being killed sexism? If not, why is clothing sexist but killing one gender not? ect. ect. I think the point behind this part of Vivian is claiming your subjective take on a game as morally superior and/or objectively true is fundamentally flawed and to most people irrelevant.
The mean to men bit is likely a specific response to a niche and unfortunately hateful group of people that exist. I would rather hope no-one would be mean to anyone and avoid whining rather than asking for help / fixing problems themselves.
I don't see how your 3 bullet points are sexist. Basically, games can be whatever you like, but the majority play them for fun. Sexism is subjective/nebulous/inconsistently applied, don't claim it as objective criticism or morally superior fact. Be nice to everyone and if you have a problem, ask for help or deal with it yourself. (?)
I don't see her as being submissive - in every post I've seen her in, she's been entirely dominant and a focus of the post. Even if she was submissive, which she doesn't seem to be, there'd be nothing wrong with that. I think you're conflating one 'mascot' as far more important than she is. I sincerely doubt anyone - GG or anti-gamer or whatever else, genuinely believes there is one type of woman or one ideal woman - but I'm sure each individual does have their preferences. I find it unlikely that the entirety of any movement could share such an individual preference, though.
Oh and also - Neither side has a monopoly on gender, this mascot doesn't just represent men, but also women and LGBT individuals too. GG doesn't have any kind of unanimous attitude towards female gamers, atleast from what I've seen, by and large they're accepting of most things as long as it fits their narrative. [The same can be said of both sides, though.]