Doc may have died in a hospital bed, but it was not in a hospital. He in fact died at the Glenwood Hotel. The "Hollywood fraud Tombstone" indeed erroneously shows Doc dying at the Sanitarium (which wouldn't exist until 1908).
According to the old west expert (TaintedTex), "he was only by kinship to Wyatt Earp a good man." Consider this: "I am a friend of Doc Holliday because when I was city marshal of Dodge City, Kansas, he came to my rescue and saved my life when I was surrounded by desperadoes" -Wyatt Earp
Actually, in his lifetime, Doc sustained more than just a minor scrape. On July 4, 1877, in Breckenridge, Texas John Henry was shot and seriously wounded. Doc had caned Henry Kahn (the other party) with a walking stick earlier in the day. The Dallas Morning Herald actually made a false report that Holliday had died, and his family was notified. Both men were arrested and fined.