While I know exactly the kind of guy you're talking about and I agree that custody issues are heavily biased in the favor of women, there are some situations where men's complaints do seem to be dismissed out of hand. For example, female on male rape, men who have been trafficked and even coerced into sex work.
I don't identify as a "men's rights" guy, but I do think that there is a problem of women accusing men of rape when in fact it did not happen. I am not saying that this is a common problem, or that it is an acceptable conclusion for police or others to jump to when hearing about the possibility of a rape. But it does happen. Rape is such a huge problem in general precisely because it's often so hard to concretely prove exactly what happened and I have certainly seen people try to take advantage of that. That said, that example of that ad campaign (don't be that girl) is awful and undermines the idea that a man can be falsely accused of rape by a woman who regrets their encounter.
I don't know if you've ever seen this Dave Chapelle bit. Granted, it's supposed to be a segue into a Clinton/Lewinski joke, but that part really stuck with me because I've seen how true that can be. Even if no legal action can be taken in a situation like that, it can result in someone being heavily ostracized in their community with no way to prove their innocence.
From what I've seen about this "men's rights" thing, it really seems to harm more than help, if only by pointing fingers and reframing serious issues in such a way as to make them look asinine.