Though the time span of this story is very short, and the setting very specific (both the time and place), it is clear that Saul Bellows is speaking a message through Seize the Day which is meant to be universal.
The main character, Tommy Wilhelm, is not so different from middle aged Americans who live and exist today. He is described as middle aged, a bit overweight, mostly lazy, mostly lonely, estranged from his family among others, and for much of the story, mostly ignorant as to why he is so unhappy. Essentially, Bellows could be talking about at least half of the nation today. The author's main point, in using such a pitiful character in a society (which wasn't terribly different from America at the time the story was published), is to demonstrate some key characteristics of human nature. These include the search for significance (in a world which is almost too large for one person to make a noticeable impact), the need to be loved and needed by (connected to) others on a personal level, and the balance of these two very different pursuits. The search for "success" (the "American Dream") is largely a personal endeavor, one that is so highly competitive that an individual must journey it alone. On the other hand, the pursuit of success is a more lonely life.
Bellow's modern view is expressed in the theme of alienation from self and others. Wilhelm is unable to sympathize with or even offer a personal connection to the elderly men all around him, many of whom remind him of his own father. Meanwhile, he is estranged from his father and desperately craving a relationship with someone to replace that emotional void in his life. The irony of Wilhelm's inability to separate his own problems from the problems of those around him is essentially the message Bellows wishes to proclaim. It is as if he views the modern world as a place that is filled with people who are so needy and self-absorbed, that they do not know how to offer help to one another.
Certainly the message, as Bellow's view and a universal theme, is depressing. Whether this theme is universally applicable is debatable.