It's a mainstream idea now that masculinity is toxic, so it's not surprising that people are projecting that onto this.
Calling it "New Warrior Training Adventure" comes across as super patronizing. "ManKind" isn't great either. The names alone are surely responsible for much of the pushback that you're seeing.
I did a bit of digging though, trying to keep an open mind, and I was surprised to find that MKP is indeed not a radical program to feminize men.
Their FAQ is pretty informative: https://mankindproject.org/frequently-asked-questions
Their YouTube channel gives a feel for it: https://www.youtube.com/channel/UCcR01KkQCYfHbzHwBlI8eeg
Basically, it's a small nondenominational religion for woke men. The leaders are a Marine Corps officer, a therapist, and a PhD of "curriculum studies". It's not very surprising to see all the comments here about how profound it was, because it was designed to be a spiritual experience.
It looks like a cult and is certainly weird (nudity is optional), but the leaders aren't getting rich off of it, and you are allowed to leave. There are regular meetings, rituals, a set of shared values, and a drive to recruit more members. It seems like mostly dudes who want to better themselves and the world at large. A lot of it is classic self-help stuff. They advertise NWTA as "Get Ready for a 48 Hour Adventure that Can Transform Your World as a Man", for example.
I think people are right to be skeptical of these types of groups, as they're playing to an audience of disaffected men and trying to improve lives with groupthink and rituals. I'd place MKP somewhere between Scientology and Tony Robbins. Probably a powerful experience in person, but best to avoid it or you might get roped into an organization.