Certainly the story would be quite different without the element of tragedy. Critic Jonathan Rosenbaum argued, in countering comments he heard that the story's heartache reflected the reality of life, that the film would have been just as realistic if Ennis and Jack had found lasting happiness together (I'm not sure about his familiarity with the original story), but however that may be, this was not the story that the artists wished to tell.
Some gay viewers took umbrage at the tragic dimension to the film (critic David Ehrenstein, for example), suggesting that this factor reflected a pessimistic, old-fashioned view of homosexuality that insulted modern gay aspirations. I simply saw it as symptomatic of the truth of many gay lives throughout history, and the fact that the film was about homophobia did not itself make the film (or the people behind it) homophobic.
Indeed, the sympathy with which Jack and Ennis are observed (despite their deeply flawed characters), and the integrity with which their passion and love are invested, convey the deeply humanistic concerns of the artists behind this story. This was the first film I saw in which gay men were treated as rounded and complex human beings, and whose lives were honored as worthy and important. And part of being a rounded and complex person is enduring sorrow as well as fun and joy (the two elements that seem to predominate in so many contemporary gay-themed films and shows).
I think you are right, Amanda, to cite the film as a classic, and to suggest that its tragedy is an important element in making it so. A classic is something that we remember, that becomes part of the fabric of our lives and our culture, and nothing makes us remember so keenly as deeply felt emotion. The recognition of love found and lost is as profound an emotion as any in the human repertoire, and this is what Brokeback Mountain reveals to us, tenderly, passionately, and unforgettably.