NYC is a bad example because it's an extreme outlier in terms of size and density. But the metropolitan area is actually much larger than the urban area; here's a map of all the counties within the NYC metropolitan area.
It covers 8,200 square miles, just slightly less than the area of New Jersey.
Metropolitan areas are quite large and typically include the core city along with the entire surrounding area that is economically and culturally heavily linked with the core city.
https://en.m.wikipedia.org/wiki/Metropolitan_statistical_area
Here is a map of all the metropolitan and micropolitan statistical areas (MSAs) in the US. Micropolitan areas are essentially defined the same way, except the core urban area of a micropolitan area is <50,000 population, while a metropolitan area has a core city of >50,000 population.
You can see that metropolitan areas include vast areas that are not even remotely urban. Beyond that, there is also a category called the Combined Statistical Area, which often combine multiple metropolitan areas.
Here is a map of LA where the red areas are urban areas, the beige counties are part of the Metropolitan Area, and the yellow counties are part of the CSA.
The CSA for LA is a whopping 34,000 square miles, or slightly larger than the island of Ireland or the state of South Carolina. However, it only contains about 2,300 square miles of urban area. Estimating the urban area is even more of an imperfect science than the metropolitan area, so I'm not sure how they calculated that number.
When people say Greater Boston or Greater Toronto, they are usually referring to the MSA, but might also be referring to the CSA. So the short answer to OP's question is that "Greater" and "Metropolitan" are roughly synonymous. FWIW, I think that metropolitan areas used to be significantly smaller and more urbanized, but they had to modify the definition over time due to trends of suburbanization and decentralization in American city development.
I don't think anybody is expecting women to do administrative work on behalf of men's rights. It's more that women tend to react with outright hostility when men advocate for other men.
It's actually the feminists who frequently argue that men need to be fighting for women's issues. I haven't seen the reverse from male advocates, partially because it's quite obvious that such a request would be summarily denied. Men generally just want:
funding for men's shelters
sympathy & aid for male victims of domestic violence and sexual assault
solutions for the growing educational achievement gap
a discussion about various legal discrepancies when it comes to conscription, marriage, and parental rights and responsibilities.
None of this requires women to assist or flex their institutional power. But when men are systematically denied access and funding for various forms of governmental aid, it seems like certain women are flexing some of their institutional power to prevent men from having access to the same kind of social safety net that women enjoy.
This is a faulty analogy, because men's issues are women's issues and vice versa. It's impossible for women to actually solve their own problems without also solving men's problems. How are women ever going to keep their oxygen masks on if they are surrounded by men who are suffocating and trying to rip the mask from their face? In order to help anyone, you need to help everyone.