I think you're totally right, they should work together. Egalitarianism is the best way.
Today there is pretty much exclusively patriarchy, under which, everyone suffers. Every gender faces oppression under patriarchy, females, males, and especially those who do not identify within the binary. I'm a woman and have seen a lot of female oppression in my society. But I have also seen a lot of male oppression as well. Both are equally unfortunate.
Extreme feminism (I don't want to use the term "radical" as that has a connotation that does not match the word's true meaning, a radical feminist is not necessarily extremist) is mostly a creation from media propaganda aimed at weakening the women's right's movements after it gained a lot of power during the second wave. Most feminists I know are well aware that men suffer too, and that there is no point trying to tear the other down so we can build ourselves up.
MRA's seem to me to be the more extreme group here, and are acting also as backlash to gains in women's rights. But they do have some valid points, men's rights should be protected as well as women's.
Male oppression as I have seen it includes the fact that men and boys do get raped and sexually assaulted, but oftentimes are even more scared than female victims to report it. I personally hate when I see statements about rape victims/rapists and abuse victim/abuser that use gendered pronouns, because it's severely limiting. A man can be abused or raped by a woman, and gendered statements that use "he/him" to describe the abuser and "she/her" to describe the victim will only serve to make him more unlikely to seek help.
Patriarchy also acts to make men feel ashamed for possessing characteristics that are deemed feminine. This is where a lot of homophobia/biphobia/transphobia comes in, as men that fall into those categories are seen as betraying the male gender stereotype.
Men can have body image issues just like women. Men are inaccurately portrayed by the media just as women are.
But, the thing that MRA's don't seem to understand, is that male oppression stems from the oppression of women. Because women are the subordinate group, for men--the superior group--to act feminine, is to become subordinate, and therefore deserve backlash by those trying to enforce patriarchy. Patriarchy starts by oppressing women, and then oppresses men to prevent them from acting like women. This is why we have such rigid gender binary, and this confuses a lot of people. Differences between genders are perceived and learned, besides a few organs and hormones, men and women are much the same; gender is fluid. We are taught from very young ages to fit into a gender box so that patriarchy can continue to control us.
There is a lot of backlash against the word "feminism". Maybe "humanism" is a better word, but feminism is a historical term and will not go away any time soon. It does not mean that feminism is all about women or that only women can be feminists. People should not be confused: feminism is about equal rights only. To get rid of male oppression, we need to eradicate female oppression and work towards equality.
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