It's true that the appeal of plays like The Great God Brown and Mourning Becomes Electra will never be as universal as that of O'Neill's family drama, partly because of its amazing craft, but partly and maybe mostly because it is a family drama. As one of the commentators says, whatever kind of family you have, you can still see yourself in it: cataclysmic as the lives of the Tyrones are, they are also the lives of a father and a mother, a husband and a wife, and sons and brothers. Long Day's Journey got a head-start on "lasting" fame (at this writing, 50 years and counting) in part because it was written -- we must assume unconsciously -- in a form that would become familiar to and beloved of all Americans: that of a TV sitcom. If the language and emotions are a little elevated for modern audiences, they can still relate to the arguments between Archie and Meathead -- I mean Tyrone and Jamie.
Most of O'Neill's other plays are much harder to get to. They are conscious (not to say self-conscious) attempts to recreate ancient tragic forms in American vernacular. To enjoy them you have to have some taste for the declamatory, the outsize, and the outrageously ambitious. In a way I like them for the same reason I like Sam Fuller and Oliver Stone.
O'Neill's greatest virtue as a writer is his passionate sincerity; his dialogue may be clumsy, his themes may be obvious, but you always get the feeling that he means it, that each line comes out of conviction rather than calculation. There were other playwrights who were more sophisticated in their attempts to re-create old forms, like Maxwell Anderson, who tried to apply the Shakespearean blank verse tragedy form to the modern-day underworld in Winterset. But the marginal sophistication of these efforts just makes them seem more dated: you get the feeling the author should have known better. O'Neill's work may be primitive at times, but it doesn't exactly feel dated, because it seems to belong less to a particular time and place than to a particular personality -- O'Neill's. And because he's so direct, so passionate and so unrelenting in setting out his themes (including, in The Iceman Cometh, just adopting the catchprhase "pipe dream" and repeating it nine thousand times), he doesn't have the problem of most English-language playwrights of his time, of being too clever to really produce audience involvement. O'Neill isn't clever (his writing isn't, I mean) and while he knew every trick of the theatrical trade, his plays are not the facile "well-made" plays that most of his contemporaries were writing. He's going to tell you what matters to him, and he's going to tell it to you over and over until you realize that he really means it. The experience of an O'Neill play is a bit like Conrad L. Osborne's description of the experience of an early Verdi opera: you start off being bemused by the crudity, but by refusing to back down, "He forces you to play on his own terms, where he always wins."
By the way, the best parody of O'Neill is probably still George S. Kaufman and Morrie Ryskind's spoof of Strange Interlude in the famous Groucho Marx monologue from Animal Crackers:
Living with your folks. Living with your folks. The beginning of the end. Drab dead yesterdays shutting out beautiful tomorrows. Hideous, stumbling footsteps creaking along the misty corridors of time. And in those corridors I see figures, strange figures, weird figures, Steel 186, Anaconda 74, American Cane 138...