To watch this indulgent but madly watchable documentary about the life and times of Liza Minnelli is like snorting a pound of uncut showbiz glitter through a rolled-up copy of Variety off Joel Grey’s naked back on the Studio 54 dancefloor – though as ever with documentaries about celebrities facing the destructive power of drink and drugs, there is no mention of the limelight and praise addiction which they are expected to maintain.

I was sorry that Minnelli’s marvellous, underrated film New York, New York with Robert De Niro is passed over relatively quickly – conveying the wrong impression that, aside from the iconic song, it’s a blip on her CV – and sorry also that her late-masterpiece comic performance on TV’s Arrested Development gets hardly a mention. But otherwise this is a richly sympathetic and thoroughly enjoyable portrait of an authentic queen of American musical theatre and movies; there is some wonderful modern-day interview footage of Minnelli, talking with waspish candour about herself, and apart from a slight vocal tremor, very robust. There is a great moment when, after having a FaceTime conversation with Mia Farrow, Minnelli is shown looking sharply at her own face in the little box in a corner of the screen: she instinctively frowns, pouts, assessing herself.

That voice, with its unmistakable little gulp, or chuckle or suppressed sob that surfaces in the middle of an extended musical line, emerges as an extension of the way she talks with the media and – as far it’s possible to see this – in private. She got this weaponised vulnerability and superpower-fragility, you must assume, from her troubled mother, Judy Garland. We see the famous (or notorious) moment when they appeared together at the London Palladium and Judy started grabbing Liza’s microphone, pushing it closer to her mouth, suddenly aware of competition, wanting to school her on stage – or embarrass her. The imperious and shrewd sense of how things are going to play on camera no doubt comes from her late father Vincente Minnelli, who also showed her Louise Brooks’s hairstyle just before she did Cabaret, a look that Liza adopted for the rest of her life.

From her unofficial godmother, the writer and dancer Kay Thompson, Minnelli learned the never-say-die ethos of the show going on, from Charles Aznavour she learned to dramatise the grit, the sorrow, the interior melancholy of a song. From designer Roy Halston she got the clothes. Then Broadway legend Bob Fosse and composer and lyricist John Kander and Fred Ebb gave her the role of a lifetime in Cabaret’s brilliant, sexy and thrillingly damaged survivor Sally Bowles, a persona which she was able to modify and reproduce in various forms for the rest of her career.

And what of the un-hilarious tragicomedy of Minnelli’s marriages? The star herself gets this film’s biggest laugh with her wearied response: “Give me a gay break …” Her first husband was singer-songwriter Peter Allen (“She was devastated when she found Peter in a compromising situation with another man”); her second was producer Jack Haley Jr, son of Jack Haley from The Wizard of Oz (“Dorothy’s daughter marries the Tin Man’s son!”); her third was a carpenter and sculptor Mark Gero, about whom we learn nothing other than his civilian status, and the fourth was the manipulative David Gest, who was not candid about his gay existence. (Someone has to write a musical about these four men: The Four Husbands of Liza Minnelli.) There is an awful poignancy in Minnelli’s attempts to have a baby, which were heartbreakingly unsuccessful.

Like Garland, Minnelli has a gay fanbase which is passionate in its emotional connection and connoisseurship – and perhaps coming to terms with this, and compartmentalising it alongside her own heterosexual identity, is something which Garland actually managed rather better than her daughter. Otherwise she had doomed relationships with Desi Arnaz Jr, in the face of opposition from his mother, Lucille Ball, and with Peter Sellers – all amazingly unworkable situations, like the six impossible things that the White Queen could believe before breakfast. Perhaps, in the end, Liza Minnelli’s authentic relationship was with the audience.

• Liza: A Truly Terrific Absolutely True Story is on digital platforms from 16 June.