I would have to say that the tone of Richard Edward Connell's "The Most Dangerous Game" is best described as melodramatic. The American Heritage Dictionary defines melodrama as a drama, such as a play, film, or television program, characterized by exaggerated emotions, stereotypical characters, and interpersonal conflicts.
One nearly universal characteristic of melodrama is the happy ending. No matter what dangers, what trials and tribulations, the hero or heroine go through, nothing very bad usually happens to them.
Both Zaroff and Rainsford can be described as "stereotypical characters." Zaroff is a super-villain with implausible motives. Rainsford is a super-hero. He even smokes a pipe and must have a handsome profile. Zaroff's emotions are exaggerated because he gets great thrills out of pursuing and killing human beings. Rainsford's emotions are exaggerated because he is in one predicament after another, beginning when he falls off a boat (while trying to grab his pipe) into shark-infested waters.
While Rainsford is swimming toward the mysterious island
A certain coolheadedness had come to him; it was not the first time he had been in a tight place.
What makes this story popular in classrooms is that it appeals to young readers while demonstrating some of the standard techniques of storytelling, including creation of a setting, creation of a conflict between a highly motivated protagonist and and an equally highly motivated antagonist, and the building of dramatic suspense. It is a very good job of professional commercial writing. It has excellent cinematic potential and has been adapted to film several times.