It is impossible to know what JD Salinger did to celebrate his 90th birthday a few days ago on January 1, but if he was anywhere near the internet, he could not have failed to notice the many tributes he has inspired. The quality of the tributes, however, leave much to be desired. Even the good ones cannot resist treating Salinger as the perpetually youthful voice of innocence and alienation. It is like Salinger the writer has simply morphed into an adult version of Holden Caulfield.

Salinger deserves better. Catcher in the Rye, like Huckleberry Finn, is a breakthrough book, and its ability to capture an era should be honoured. But at the heart of Salinger's major work – his fictional history of the Glass family – is not lightness, but the observation of French novelist Albert Camus, who wrote in his essay The Myth of Sisyphus that "there is but one truly serious philosophical problem, and that is suicide. Judging whether life is or is not worth living amounts to answering the fundamental question of philosophy".

In Salinger's case the philosophical problem begins with the suicide of Seymour Glass in the 1948 short story, A Perfect Day for Bananafish, and then continues through five longer stories published between 1955 and 1965, in which three of Seymour's siblings Buddy, Zooey and Franny struggle to come to terms with their eldest brother's death and their own lives.

Seymour's suicide is particularly grim. It occurs during his honeymoon in Florida, when after a day at the beach, he lies down in the twin bed next to his sleeping wife and shoots himself with the gun he packed in his luggage. How could Seymour, the most philosophical and gentlest of the Glass children, take his own life? This is the grim question on which the Glass family stories are based and for which Salinger provides no easy answers.

Seven years elapsed between A Perfect Day for Bananafish and the next Glass family story Franny. Salinger then began writing exclusively about the Glass children and their parents, but his stories did not follow family chronology. The earliest picture we get of Seymour comes ten years after Franny in the final Glass family tale Hapworth 16, 1924. It is as if Salinger realised he was engaged in his most important work once he began to write about the Glasses, but by then it was too late to do anything systematic.

The result is that anyone reading Salinger, like Seymour's brothers and sisters, is forced to rethink the Glass family past. From the fairytale-like story Seymour tells Sybil, a young girl he meets on the beach, about bananafish who eat so many bananas that they cannot get out of the banana hole they have swum into, we can guess that Seymour found himself trapped in a bad marriage from which he could see no escape. When Seymour's brother, the writer Buddy Glass, says that "all we do our whole lives is go from one little piece of Holy Ground to the next", we can imagine that Seymour reached a point where he believed that he had run out of holy ground. But such conclusions are guesses at best. Salinger does not invite us to do what Holden Caulfield says he wants to do with his favourite writers Isak Dinesen and Ring Lardner – call them up on the phone for a friendly chat.

If we want to put Salinger in historical context we need to think of him like Ernest Hemingway – an American writer who was profoundly changed by war. In Salinger's case the war was the second world war, and like so many veterans of the that era, Salinger rarely talked about his battle experience. But in her memoir Dream Catcher, Salinger's daughter Margaret remembers her father telling her, "You never really get the smell of burning flesh out of your nose entirely".

Small wonder that Salinger made such a remark about his wartime service, which ended when he was hospitalised with battle fatigue. Drafted in 1942, Salinger served in the 12th Infantry Regiment, which landed at Utah Beach on D-Day and then fought in the Hrtgen Forest in Luxembourg and in the Battle of the Bulge in Belgium before coming onto newly liberated Nazi concentration camps.

In A Perfect Day for Bananafish, there is brief mention of Seymour's service in Germany, but it is in Salinger's 1950 short story, For Esmé – with Love and Squalor, that we get the fullest picture of the war's impact on him. In For Esmé, the war is central to the thinking of the story's narrator, Sergeant X, who is a counter-intelligence officer like Salinger. At the end of the story, Sergeant X is in Germany recovering from a breakdown when he receives a package from Esmé, a young girl he met one afternoon at a restaurant in Devon, England. Enclosed in the package is the wristwatch of Esmé's father, a British soldier killed in Africa. It is a magical moment. The gift and the memory of Esmé are enough to make the Sergeant X think that he may become a man "with all his f-a-c-u-l-t-i-e-s intact" once again. Esmé has done for Sergeant X what Sybil could not do for Seymour.

But for Sergeant X and all the Glass children, staying sane remains a struggle. Sanity is never a given, which is why the rosy picture of Salinger on his 90th birthday undermines his life experiences. It leaves out the darkness that fuels his work and makes his comedy so hard won.