Yes, the story still seems relevant today. It would probably have seemed relevant thousands of years ago, and I suspect that it will remain relevant as long as human beings (young men in particular) are constituted as they are. The desire for respect is probably universal, and the unfortunate consequences of efforts to win that respect in superficial ways have probably always been with us and will probably always remain with us. Wright's story is partly a story of growing up, of learning by making mistakes and even of learning by making foolish, laughable mistakes -- experiences that are probably inevitable aspects of the human condition.
While the desire to be grown--indeed, the feeling you are grown--is immortal, witness Romeo and Juliet and all of Romeo's friends, something is happening in our era that is very different yet reminiscent of Dave's experience. Dave had a fascination with, you might almost say a fixation with, owning a gun presumably because of some exposure to guns. In our era, the fascination with dangerous objects, acts, ways of life, and associates is ramped up, heightened, because the exposure of children and young people to these things is ramped up and heightened. Exposures come through music, movies, television, books, news stories. The result of these fascinations is similar to Dave's: he desperately wants the trappings, the accoutrements, of adulthood without first gaining the wisdom and responsibility of adulthood and, it seems, so do young people today--one difference is, it is all so much more easily within reach today.
The desire to reach adulthood is concomitant to a desire to gain some respect. If a man hold a gun, he will get the respect that comes from fear. Dave understands respect in this denotation, therefore, he desires a gun. Then he will not hear people say, "You ain't nothing but a boy."
But, for many youths nowadays, there is no need for something like a gun since they receive respect gratuitously from their parents and even from some of their teachers and others in the community--all those who have embraced the new ways of society. As the poster above calls them, the "mainstream" feel no need to prove themselves as all is already given them.
I do think that essentially all young people today (and when I was young) want desperately to become adults. However, I think that the particular way in which Dave wants to become an adult is a little less common in mainstream US society than in some others.
In cultures like the one where I grew up, being a man is very much connected to being capable of violence and being able to exert power over other people. I think that young men growin up in societies like that tend to feel urges like Dave's -- they want to have that gun. More specifically, they want and need to feel powerful.
So, I think that the desire to become an adult is universal, but Dave's method of doing so is less compelling to most American "mainstream" youths and more compelling to those from more "macho" cultures.
I am reminded of another short story called "Through the Tunnel" by D. Lessing, which is about a teenage boy who is desperate to grow up, and designs his own initiation ceremony so that he can feel that he has passed "through the tunnel" from childhood to adulthood. I think that a desire to be recognised and accepted as an adult by society is common to all teens, and this story draws attention to the way that such youths as Dave are often forced to try and be accepted as a "man" by drawing upon illegal means.
To a large extent, I would say that one of the factors involved with our current rates of gun violence is similar to the experience Dave has with his gun. Simply put, there are many reasons why young people use guns in a reckless manner. One of them is the perceived belief that it makes them feel "like a man" or older when they use a gun. Dave's desire to feel power in a world where he lacks it can be seen in the modern experiences of kids who use guns today. For many situations with today's examples of young people and gun violence, it can be traced to situations where, like Dave, they too wish to feel a sense of power. Dave's desire to "become a man" is rooted in the fact that he wishes to feel and experience power. In a social setting where he does not get a chance to experience this, the ability to use a gun, to find a quick fix to this systemic problem, is where disaster is both apparent and recurring, as he wishes to take the gun with him to the North. The reality is that for young people today who, like Dave, lack power over their own lives, the ability to use guns and weapons as a way to mask this lack of power into believing they have it is still there.