The main thing that brings the Red Ajah together is hatred of men

Untrue we meet several among the red who don't hate men, one likes to play fairy godmother to her all her nephews and their descendants, another likes slumming in the tar valon taverns and grinding up on men and dancing on the bartop. Many joined the red solely because they think that the red ajah is saving the world and protecting it from dark friends and men who would break the world. There will still be dark friends and dreadlords and they still need to meet and deal with male channelers. How they deal with them is what will change.

Many of the other ajahs have more to deal with or risk irrelevance. We see that with the grey starting to really study traveling to update what they do. The yellow nice as it is needs to decentralize from the tower to stay relevant or the kin and wise ones could take their place since the kin are already everywhere and that's now the wise one's job to be everywhere as they enforce the treaty. The brown needs to distribute the knowledge they hoard. The green needs a new purpose since theirs was the last battle. The white never really did much. The blue is sort of ok because they're already ill defined and just try to fix things I guess.

There are many men haters among them. But they don't make up the entire ajah.

And in the books we've seen the red among all the ajahs actually work to find relevance in the changing dynamics of the world.