Like nature’s annual flying ant day, Radiohead’s infrequent live appearances in London draw little-seen wildlife out into the open. In the crowd, amid the refurbished steam-punk splendour of the Roundhouse tonight, are the like-minded souls Kate Bush, Nick Cave and P J Harvey – artists whose work, while not precisely akin to Radiohead’s ebbing and flowing machine dreams, also mixes the artistically uncompromising with the commercially successful. Also negotiating an indie T-shirted fortysomething queue that snakes south towards Camden Town are the former Doctor Who Matt Smith, the TV comedy subversives Adam Buxton, Chris Morris and Julian Barratt, and the actor Toby Jones, a former schoolmate of the band.

This second of three Roundhouse shows is one of those rare evenings that feels like an event – an unofficial celebration of the notion that, even in a music world increasingly denatured by rote professionalism and the tyranny of the quantifiable digital metric, outsiders and weirdos can still win.

It’s an excitement also born of the hyper-compressed timelines now necessary to capture our collective attention. Three weeks before the Roundhouse shows, there was only the barest indication that Radiohead were about to release any new music at all. Then, in less than a week, they erased their website and social media presence, put out two startling singles – “Burn the Witch” and “Daydreaming” – and lavish videos on consecutive days, then released a new album, A Moon Shaped Pool, at less than 24 hours’ notice.

This is how it goes when music must operate as a subset of the social media industries. Songs and images that you may have worked on for years are fired into the public consciousness over a period of days. Yet, with A Moon Shaped Pool, Radiohead have a record – subtle, insinuating, emotionally rich, endlessly replayable, a career highlight – that is well suited to discovery by unconventional means. Although its melancholic tone is rooted in the disintegration of the frontman Thom Yorke’s relationship with his partner of 23 years, the surprise is that the album is a hopeful thing. Yorke’s lifelong concern, gnawing anxiety, is alchemised by an empathic band and Jonny Greenwood’s increasingly expressive string arrangements into something beautiful and capable of being mastered.

Whatever else has happened to Radiohead, it’s agreeing with them. Tonight’s show is exultant: a vivid, high-spirited and even genial performance of songs that many think are inaccessible or introspective to the point of misanthropy.

The intro tape features the laughter of Nina Simone, possibly the least Radiohead sound imaginable; the first song, “Burn the Witch”, augments the recorded version’s nervous strings with a heavier dose of Colin Greenwood’s bass and, in the stage lighting, a flaming vision of hell on Earth.

Yorke’s voice is a Dylanesque croak at first but soon finds its purchase. He darts back and forth across the Roundhouse stage for all the world as if he’s an actual entertainer. Radiohead front-load their show with A Moon Shaped Pool’s first five songs, which makes for an unfamiliar beginning, but these intricate studio compositions take on new life with the boom and resonance of a big room. Then they play the bravura tantrum “Airbag”, the beatboxing dub-rock opener from 1997’s OK Computer, which reformatted Radiohead as a band whose job it is to change rock every four or five years – and the roof, as they say, comes off.

Though never anyone’s idea of lumpen singalong indie rock for the lads, the notoriously reserved Radiohead do have their communal moments. “No Surprises”, their lament for a stymied suburban Mr Clean, is redeemed from condescension by a delicacy of touch that remains intact even when 2,000 people are singing it. The closing Beatles-ish “Karma Police” gives way to a lairy, open-ended “Hey Jude” chant of its refrain: “For a minute there, I lost myself.” These are relics of Radiohead’s forgotten years, when they wrote conventional songs rather than working obsessively to perfect textured swatches of feeling. Yet the emotional connection remains undeniable. For many people, Radiohead speak for them even when they’re not quite sure what Yorke is saying.

There are moments of humour, too. In his best public-service announcement voice, Yorke says that the posh seats will be electrified in 20 minutes, so please stand up and dance. After “No Surprises”, he affects a Donald Trump face, uncharacteristically milking the applause with jutting chin and comedy Mussolini-style head-nods. When an over-enthusiastic fan punctures a quiet moment by yelling, “We f***ing love you, Thom,” he responds not with a basilisk glare but a shy smile and a muttered: “Uh, yeah, you, too.”

It’s all surprisingly sweet, an experience of warmth and companionship with the band you’d least expect to provide it. Some 25 years into their career, Radiohead are at the top of a game no one else even knows how to play. Difficult music has never been so easy to listen to.