There are unofficial rules about how a modern biography should begin. To start with the birth or even the death feels increasingly generic and stale. Instead, you are expected to find an incident from the middle years, something dramatic and specific like a lost manuscript, a duel, an accident that leaves your protagonist radically reconfigured. It’s got to be snappy and tense, the sort of thing that makes readers feel that they have landed in the middle of a particularly thrilling heist movie in which it’s unclear whether anyone will get out alive.

So when Sue Prideaux decides to open this life of Nietzsche with something else entirely, you know you are in the hands of a biographer who is either incompetent (unlikely, given that her earlier books on Strindberg and Munch have scooped prizes) or very sure of herself. For instead of showing us middle-aged Nietzsche making horrible animal noises while his prim mother receives visitors in the sitting room, or peak Nietzsche striding over the Alps imagining himself as an Übermensch (usually translated as “Superman”, even though it sounds silly), she gives us indeterminate Nietzsche, Nietzsche before he – or anyone else – has much idea who he really is.

Prideaux’s narrative opens with a lengthy quotation from a letter the young man wrote on 9 November 1868 while still a student of philology, a subject he didn’t much care for, at the University of Leipzig. True, he is describing a highly significant day – the one on which he first met Richard Wagner, the composer who would become his mentor and inspiration. But the 24-year-old also packs in a great deal of extraneous detail that Prideaux insists on quoting in full: the loss of a smart new suit, an impromptu tutorial on the Eleatics (a Greek school of philosophy), sheets of sleety rain.

Here is Nietzsche as most of us have not encountered him before: self-deprecating (he loses his trousers and finds it funny), unpredictable and, above all, sociable – friends arrive and leave, he dribbles away time in a popular student restaurant before finally gearing up to meet the great man. Even more remarkably, Prideaux resists the temptation to editorialise, to tug nervously at our sleeve to make sure we’ve got the point that Nietzsche is so much more than the sinister pin-up of mid-century fascists and serial killers (Moors murderer Ian Brady was a fan). He was actually quite fun and, dare one say it, normal.

None of this will come as a revelation to academic philosophers who have watched over the rehabilitation of Nietzsche for nearly a quarter of a century. Until the 1970s the Prussian pastor’s son was often carelessly characterised as a Nazi sympathiser, despite having died in 1900. It is easy to see how this notion arose: Nietzsche’s ideas about the Übermensch could easily be repurposed as an argument for Aryan supremacy, while the title of his posthumously published The Will to Power seems to demand a soundtrack of marching jackboots. Over the years, though, reparative readings of Nietzsche’s work have begun to allow the original thinker to emerge. These days Nietzsche is even viewed as a postmodernist visionary: “There are no facts, only interpretations,” was one of his famous declarations, along with “God is dead”. In the last book he wrote before mental illness descended he set out a series of immodest proposals: “Why I Am So Clever”, “Why I Write Such Good Books” and, in case anyone was wondering, “I am not a man, I am dynamite”. You could almost call him a performance poet.

Anyway, Prideaux tells us repeatedly, the Nazi high-ups never really claimed Nietzsche as one of their own. Ernst Krieck, a prominent ideologue of the Third Reich, once sarcastically remarked that, apart from the fact that Nietzsche was neither a socialist, nor a nationalist and was opposed to racial thinking, he might indeed have been a leading National Socialist thinker. Hitler, who was neither as clever nor as funny as Krieck, once told Leni Riefenstahl: “I can’t really do much with Nietzsche ... he is not my guide.” In fact, there is no evidence that the Führer ever read him. The one political extremist happy to acknowledge his influence was Leon Trotsky, who liked to imagine the Übermensch as a sweaty Soviet factory worker.

In other areas of her subject’s life Prideaux feels under less pressure to offer persuasive revisions of stock narratives. There’s no gainsaying the fact, for instance, that Nietzsche remained weird around women. With a fair degree of tart wit, Prideaux retells the story of his infatuation with Cosima Wagner. Nothing odd about that – Richard Wagner’s second wife was a compelling presence – until you learn that while Nietzsche was sighing over his lady he was also running all over town on an errand to locate the silk underpants without which the composer of Die Meistersinger felt himself unable to write another note. And then there is Lou Salomé, the wild Russian femme fatale who may or may not have padded her bodice (the whole of intellectual Europe speculated wildly about her unfeasibly large bust). For three years Nietzsche lived with Lou and his gay best friend, Paul Rée, in a stormy philosophico-erotic triangle that involved no sex but a lot of very fine Alpine views.

The trickiest of all Nietzsche’s women was his younger sister, Elisabeth Förster-Nietzsche. Having returned to Germany in 1890 after a failed attempt to establish an Aryan colony in Paraguay with her notoriously antisemitic husband, Elisabeth spotted a new chance to make history go the way she wanted. With Friedrich now ailing (Prideaux is agnostic on whether syphilis was to blame), she seized control of her brother’s literary estate and spent the next 40 years editing his major texts and private letters until he started to sound remarkably like her, which is to say crudely nationalistic and racist. Under her vigorous, capable and corrupt direction, the Nietzsche Archive became the playground of rightwing propagandists who took the view that Nietzsche’s published works hardly mattered because the real meat of his work was to be found in the Nachlass, the unpublished literary estate Elisabeth had already gutted to her own ends. Indeed, it comes as a shock to rediscover that The Will to Power is not by him at all, but consists of fragments soldered together at a later date by others who truly believed they knew what he meant to say better than he did himself.

The great pleasure of Prideaux’s sprightly biography is watching philosophy in the making. Reading about Nietzsche’s life, which had as many false starts and wrong turns as anyone else’s, is to be reminded that systems of thought do not arrive unbidden in the library or the lecture hall, but are worked out in the mess of everyday life. For all his declamatory certainty, Nietzsche developed his ideas amid love affairs, lost trousers and a bullying younger sister. Academic philosophers may feel that there is not much new to detain them here. For the rest of us, this biography is nothing short of a revelation, a sort of word made flesh.

• I am Dynamite! A Life of Friedrich Nietzsche is published by Faber. To order a copy for £22 (RRP £25) go to guardianbookshop.com or call 0330 333 6846. Free UK p&p over £15, online orders only. Phone orders min p&p of £1.99.

• This article was amended on 21 November 2018, to remove a photo purportedly of Nietzsche after it was flagged as fake. After contacting Alamy, the veracity of the photo could not be confirmed and it was removed.