I’m not entirely sure there’s any word or term that wouldn’t put people on the defensive. Doesn’t matter what word we use for it. They don’t like “feminist” as a word either and it doesn’t include any mention of maleness at all. The fact is, we have a societal problem rooted in male aggression and domination.
There are 20,000 calls to the National Domestic Abuse Hotline every day. A woman is raped every 2 minutes somewhere in the USA. There are over 1800 women killed by men annually, over half by intimate partners. That’s before we even talk about getting paid less, being less safe on the streets, the glass ceiling at the office or that women still do most of the housework.
The problem is perceived pecking order. We see that everywhere, as you pointed out. Just try telling a racist that people of color are equal to them.
The people who are triggered by words that refer to a system based on gender inequality will not like any term anyone can come up with. Because they liked it the “old” way. And it’s not just men. Lots of women in that camp, too.