When Griffin Dunne announced he had the go-ahead to film a documentary about Joan Didion, the great writer (and, as it happens, his aunt), he crowdfunded the money needed in a day. Netflix kicked in with the rest and the result is The Center Will Not Hold, streaming now.

Didion has been the master of holding readers back while appearing – in her prose at least – to let you in. So fans, of which there are many, have been hoping for an unguarded moment – something that unwraps the enigma.

Those moments in the documentary are few and far between. But that a film as slight as this is still a must-watch is testament to Didion’s extraordinary and enduring appeal. New generations discover her as they discover The Catcher in the Rye. She becomes their author. Falling-apart copies of Slouching Towards Bethlehem get passed down the generations. And the thrill of discovery is not just in seeing the 1960s represented through a dark mirror, but in the prose style: long, breathless sentences that chart an inner turmoil combined with an astute eye for the times and what those times might mean.

Didion is a favourite writer of young women. Her instructions for packing (leotards and bourbon and cashmere shawls and … a typewriter) are republished in fashion magazines, and her look (ironic, sunglasses at all times, a scarf tied under her chin) has never really gone out of style. Old photographs of her look achingly chic, and in her later years, she even modelled for Céline.

Caitlin Flanagan in the Atlantic writes: “Women who encountered Joan Didion when they were young received from her a way of being female and being writers that no one else could give them. She was our Hunter Thompson, and Slouching Towards Bethlehem was our Fear and Loathing in Las Vegas.”

Part of Didion’s appeal has always been her “coolness”. That famous picture of her leaning against a Corvette was an Instagram moment decades before Instagram. But beneath this cool exterior, there is the suspicion that she was always a lot frailer than she let on. In The Center Will Not Hold, we see her as physically very frail, but still presenting as the “cool customer” who, as she recounts in The Year of Magical Thinking, didn’t cry at the hospital after her husband died. As film-maker, Dunne is protective of her: we don’t see her crack.

In writing about “serious” matters, Didion never sought to excise fashion or fabric from her prose. She started her writing career at Vogue and her descriptions of political or the personal often break to include descriptions of furnishings or curtains or clothes.

She writes in her essay, Goodbye to All That, about how “all that summer the long panels of transparent golden silk would blow out the windows and get tangled and drenched in the afternoon thunderstorms”.

The curtains are never just curtains, however – the details always mean something more. Those curtains mark the turning point where the exterior world and inner tumult meet and something must change. Similarly, the dress worn in the trial of Linda Kasabian in The White Album essay is a vehicle to describe Didion’s horror at both the banality and evil at the heart of the Charles Manson murders.

Martin Amis, when reviewing The White Album in the London Review of Books, noted “the volatile, occasionally brilliant, distinctly female contribution to the new New Journalism, diffident and imperious by turns, intimate yet categorical, self-effacingly listless and at the same time often subtly self-serving ... Miss Didion’s writing does not ‘reflect’ her moods so much as dramatise them. ‘How she feels’ has become, for the time being, how it is.”

Amis’s review reflected a long line of disdain from “serious writers” for the personal essay, the practitioners of which are mainly women.

“How she feels” did become “how it is”. The work has lasted, and although there are many imitators – and even more, like the novelist Bret Easton Ellis, are heavily influenced – Didion’s voice remains singular.

It’s her voice that is part of the appeal of her writing – at least for me. Her fragmentary style, particularly in the early collections The White Album and Slouching Towards Bethlehem, renders an event closer to a form of poetry than the blunt instrument that is the inverted pyramid of news or even more conversational-style features.

Didion’s style works because of, not in spite of, its fragments. It is closer to how we actually see the world – like the moment you heard big news, and how that moment is compromised of many things: how you felt, where you where, the people you were with, how cold it was outside. In Didion’s world, how we experience news is how we experience the world and that is why the temperature of the air and background music in the bar can seem inseparable from our experience of the event. To shear one off from the other is to only convey a part.

After Didion, it seems incomplete to ask only “what happened” and neglect “how did it feel?” (Ta-Nehisi Coates’s brilliant We Were Eight Years in Power does both to great effect.) Her strongest essay Slouching Towards Bethlehem – an exploration of the dark side of the 1960 – expresses dismay at the disorder of the times, and uses Yeats’ poem, The Second Coming, to describe a collapse of common values.

But as disorder burrowed more into Didion’s own life, with the deaths of her husband and daughter, her work and her sentences became even more fragmentary.

Her 2011 book, Blue Nights, is a book of shards – shorn-off sentences, fragments, words, repetitions, images held tight – like talismans that litter the text. Those of us who had fallen in love with the fragmentary, collage-y nature of Didion’s work were left without anything to hold those fragments together or give them shape.

This film picks up the pieces, a bit, and gives a softer, less alarming view of Didion than the one the writer chose to represent herself in her later works. There are things the film doesn’t address – darker topics, such as the deaths of her daughter and husband, and charges that she failed as a parent are left well alone.

Instead, we see Didion dwarfed by her giant refrigerator. We hear her talking about what she used to have for breakfast (always sunglasses-on in the house in the morning, always a cold Coca-Cola from the fridge), we hear about her morning routines with her daughter and husband, and see her now in cashmere and a gold chain – very thin, but very in control, telling us just enough about her life to keep us interested, yet also, perceptibly, holding back.

• The Center Cannot Hold is showing now on Netflix

• Brigid Delaney is a Guardian columnist