Both movements suffer from a collectivist us vs. them mentality. To MRAs, men are forced to risk life and limb to protect those privileged women. To feminists, women are being oppressed by the misogynist patriarchy. Things aren't actually that simple though. You can't just say that men/women are oppressed, and pin it on some ideological boogeyman. Men and women both have problems that they have to deal with. It's much more effective to look at the individual issues than to wrap everything up in a religion.
Just because the movement doesn't solve every problem or explain everything doesn't mean it's not useful. Of course things aren't as simple as men this and women that however there are still issues that can be worked through using feminism. There is plenty more to feminism beyond men/women are oppressed but you have to actually look for it.
I've heard a lot of different views on the wage gap some with more merit than others, but really there's no way to get rid of misinformation in any subject. For example science, or politics. The thing about feminism is there really is no governing body saying what's accepted as there is in some sciences. There's no "Bible" either as there would be in religions. It's really not regulated at all which I think was a good idea in thoery but really doesn't work in practical terms. It's a giant clusterfuck actually which is why I don't really get the comparison to ideologies or religion.
There really isnt even one bible. There are multiple translations and an infinite number of number of interpretation. Feminism at its core is about equality for women the same way that Christianity is about the bible. After that it's up to the individuals interpretation, does it mean mandated equal wages? Does it mean special rights for women that are in place to fight years of inequality? Does it mean that men should shut up and get back to the kitchen and have no property rights for the next 2000 years to counteract all the previous oppression . All of those could theoretically mean equally just like all sorts of shit can be interpreted from the bible.
It's impossible to look at issues alone because they all intersect. My womanhood is not just about being a woman. It's about being queer and Jewish as well and the meanings that are ascribed to me because of that.
Isn't that how ideas and movements work. You have one group of ideas that goes too far so a counter group springs up that works out to balance the first. In the same way that I assume you don't acknowledge all the ideas of modern feminism you can acknowledge some ideas of MRA and still reject others. At least in America ideas and movements are not an all or nothing deal although opposition groups always try to frame it that way.