Hi,
Well, this is an interesting little thread here... If I may, I would like to add a note or two.
To begin with, the concept of a "real man" needs clarification. The OP has an instance where a non-martial artist makes the claim that "real men (essentially) don't need martial arts", and to use such lowers the man's masculinity. For the record, I also see such misuse of the term Alpha Male in place of Real Man in this context. The people making these claims are not aware of what a Real Man actually is, instead attempting to imitate what they believe it to be (for various reasons), becoming "false" Men, or "false" Alpha Males. So let's look at what a Real Man (or Alpha Male) really is.
A true Alpha Male is driven, they know what they are after, and are willing to go after it or seek it out. He is generous to those around him, by caring for them and protecting them, even in ways that are not immediately recognised as generous (the way a caring parent will ground or scold a child, as well as protect and provide for them). He is confident and deliberate, even in his acceptance of any lack he has. He is congruent at all times. They are also internally validated, needing none from any external source.
There's obviously more, but this is a good list to start with.
When it comes to violence, an Alpha Male does what he needs to do. He doesn't necessarily seek it out, but he doesn't avoid it either. He is confident in his ability to handle such aspects of life. And in that we find how a Real Man fights. He fights when he needs to, how he needs to. He embraces the idea that violence is a part of his life, and one that he needs to be able to handle.
The most important thing in regard to a Real Man and violence is that he does actually seek to understand and deal with it, rather than hiding from it, and that involves in many cases taking up martial arts, joining the army, or some other variant. So the entire premise in the OP's story is based on a lack of understanding of a Real Man.
What an Alpha Man is not, though, is what we see in the OP. That is an example of a scared false Alpha, trying to control something that is outside of their sphere of control. They are also attempting to appear stronger than they are, which is an attempt to be externally validated.
There are a number of reasons for these false Men being prominent, including a lack of positive male role models in modern society, but the main thing to recognise is just that most guys around are simply not "real men", they are just playing at being one. It is a show of false bravado, but the danger is when they get put in a position where they feel they have to "back up" such thoughts and words. Then they can be dangerous.