What is the genre of Death of a Salesman? How is this significant to Willy Loman?

While it might be tempting to consider Death of a Salesman as a modern "English" tragedy written by an American playwright, it is more accurate to classify it as a "problem play."
English tragedies go back to the sixteenth century and are best exemplified by Shakespeare's serious dramas, such as Macbeth, Hamlet, and King Lear. English tragedies differed from classic Greek tragedies in that they allow for subplots and some interjected comedy and feature heroes who come from all stations of life, not just nobility. The English tragedy, although it offers a lesson on religion or ethics, also serves as entertainment. Tragedies have an element of catharsis for the audience; that is, they induce such a deep release of emotion that they cleanse or refresh the spirit.
Problem plays developed in the nineteenth century and were popularized by Henrik Ibsen and Bernard Shaw. Such plays deal with contemporary and controversial issues in a realistic manner. The purpose behind the works is to promote deep thought and discussion of serious topics.
Death of a Salesman fits the genre of the problem play better than the genre of tragedy. Tragedy requires the audience to develop deep empathy for the character who suffers the fall from greatness because of a tragic flaw. Willy Loman simply is not likable enough to produce the appropriate feelings in most audience members. It's not just that he's not a member of the nobility; he's not noble in character. He is so deeply flawed that most audience members will not personally identify with him.
However, they may see in Willy Loman the flaws of many fathers they know, perhaps even their own. Biff is the sympathetic character in the play, and he actually has a positive character arc that leaves the audience with hope—which is not typical of a tragedy. The subjects of Biff and Willy's relationship and the overall family dynamic stimulate deep thought and discussion—as does the issue of suicide. These contemporary issues portrayed through a realistic family scenario identify Death of a Salesman as a problem play.
https://literarydevices.net/tragedy/

https://www.britannica.com/art/problem-play

Arthur Miller's Death of a Salesman would be considered a family drama because it chronicles the numerous struggles and conflicts in the lives of the Loman family as Willy and his sons attempt to attain the "American dream." One could also categorize Death of a Salesman as a tragedy because the protagonist suffers as a result of an inherent character flaw, which results in his tragic downfall. However, Willy Loman is not a traditional tragic hero like the protagonists depicted in the ancient Greek plays. Willy Loman does not hail from a royal family, is not revered throughout society, and does not have extraordinary potential, unlike Shakespearean tragic heroes such as Hamlet, Macbeth, or Othello. In the character of Willy Loman, Arthur Miller sought to depict the "tragedy of a common man." In his doing so, audiences are able to personally relate to Willy Loman's dreams, aspirations, and failures. Miller also critiques American culture by poignantly illustrating the futility of pursuing the "American dream," which is another example of how Death of a Salesman is similar to a classic tragedy.