The most egregious of the film's many transgressions is the way it paints Keaton's relations with his studios. In the film, Keaton shows up at a studio and talks his way into a contract in which he will star and direct his own films after appearing in a small role in one film (and despite the best efforts of an obstinate director played by Peter Lorre, of all people). He then happily works at that studio ("Famous Studio", seriously it is called that) for years, making hit after hit with his silent films, only to be unable to transition to sound (because of an inability to say his lines properly) which exacerbates his drinking problem to the point that he's no longer able to function. Through all this, his wife (played by Ann Blyth) loyally and understandingly stands by him, until she doesn't. In the end, Keaton triumphantly returns to vaudeville where he finds happiness reenacting his old routines and making people laugh.
In reality, Keaton worked for quite awhile before getting a starring role, most prominently as a sidekick for Fatty Arbuckle, one of the biggest stars of the era. It was only after the scandal that killed Arbuckle's career that Keaton became a star in his own right, and even then he only occasionally was his own director, more often working in collaboration with Edward Cline or someone else. During the silent era, he didn't work for a studio, rather he worked with independent producer Joseph Schenk. It was at the end of this era, when sound came in, that Keaton signed with MGM (after the financial disaster of The General), where he made his last silents and several talkies. Far from being unable to transition, Keaton's talkies were immensely successful commercially, though fairly weak artistically, mostly because the studio system limited his creative input as much as possible and refused to let him even co-direct. (I reviewed all the late Keaton films here).
Keaton was an alcoholic, and his problem became increasingly worse during the MGM years. But all accounts I've seen place the blame for that on his awful marriage (his second) to Mae Scriven (1933-36). Both of these marriages are unmentioned in the film, though the marriage that is in the film seems roughly comparable to his third and final one (to Eleanor Norris) from 1940 until his death in 1966, that he credited with helping him kick alcohol and restart his career. A career which far from being confined to the vaudeville stage, had him steadily working in film and television for 25 years.
But does any of this matter? Does a film, even a biographical picture, have to be bound by the basic facts of history? Shouldn't it be allowed to tell its own story in its own way, as long as that story is itself, entertainingly told? I do think there's a limit beyond which the truth can be stretched too far in the name of art. Despite our best efforts to ignore, obfuscate, reframe, mythologize and narrativize them, things actually did happen in the past just as they continue to happen in the present and will keep on happening in the future. There really was a Buster Keaton.