A lot of people like to complain about hate and misogyny among incels, MGTOW, etc.

I'd question how big of a problem it is compared to how big of a problem some people present it as, but there are a certain number of men who become disillusioned towards women as a result of the disadvantages that they face as men. The red pill movement even has a name for this: it's called the "red pill rage" period. And it's caused by men slowly waking up and realizing how much they've been lied to about privilege, and how disadvantaged they really are (especially when it comes to things like dating and romance).

I imagine that quite a bit of misogyny, both today, and in the past, can be traced to this. And I think there would be far fewer men going down this path if society was more fair to them.

For example, many MGTOW types complain about women using men for money. This comes up a lot in the context of marriage, where our current laws and social norms actively encourage this behavior.

If women could no longer use marriage as a mechanism to steal wealth and income from men, there would be fewer men going through nasty divorces and then identifying women as the cause of their suffering.

By advocating for better divorce and child custody laws, we are addressing some of the underlying factors that cause this kind of hatred and misogyny to begin with.

The MRM can also help contextualize the suffering of men who are victimized by the system, thus reducing the anger that they feel and their propensity to transform that anger into blind misogyny (this short Bloomberg video about the International Conference on Men's Issues has an example of this).

Therefore I think the MRM is ultimately working to fight against the "incel problem" and misogyny in general.