Why is the soldier not a man every bit as much as the leader who ships him of to die?

They are, I don't think you'll ever see feminists gatekeeping men telling them that not being a leader is not being a man. I feel like this question is very disingenuous. I doubt anyone would ever say a soldier is not a man, even by people who do gatekeep men and question peoples masculinity.

More generally, why are the men at the bottom not men?

See above no feminist is saying that men who face hardships or live hard lives are not men.

Why do only the men at the top count when you evaluate which group is privileged?

Privilege is not going to define your life. Privilege does not define other parts of your life. For example, I am white, I am also a female student on welfare and working while I'm studying. Just because I'm white doesn't mean I don't have problems. What it does mean is that I don't have to deal with a whole other set of problems that other races face.

With regards to looking at the top and looking at power, that is kind of separate. Yes, people who are more privileged are more likely to be able to accumulate power. Like if they have class privilege, racial privilege or gender privilege etc. However, the reason feminists frequently look at power is because half of the population are very under-represented in power. This is what we are talking about when we talk about patriarchy.

Patriarchy is when the people in power are men, it does not mean that all men have power. It means that the people who are making policy and making decisions for large groups of people, are men.

Has it got something to do with the more general tendency of women to ignore all men except the very few at the top?

No, it's just that the discussion about power or patriarchy is separate from the discussions about individuals. One discusses a system, the other is about people. So leaving aside power, looking at an individual level things like 'no fault divorce', birth control, objectification are things that feminists look at that affect individual people.

As an aside, feminism is a women's movement. So it's not a great framework for discussing mens issues, however I know that a lot of people here do care about mens issues, and for that I would check out /r/menslib.

Personally, on top of feminism I advocate for mens mental health programs, the Australian Mens Sheds Association and I try to actively talk to the men in my life about how seeking help for mental health issues does not make you any less manly. These are things that are hugely impactful for "men at the bottom".