Blessed are the musicmakers. Back in 1970, Andrew Lloyd Webber and Tim Rice put the guitar, the soul and the hippy spirit into the Passion and, almost half a century on, their songs are still bona fide belters. They aren't just the core of Jesus Christ Superstar; they are its sum total – it's first, it's last, it's everything.

Originally released as a concept album, Jesus Christ Superstar debuted with a lavish and outlandish American staging – one Lloyd Webber would later declare "a vulgar travesty." Tim Sheader's resurrection – Superstar's first thought-through theatrical outing in 20 years – wisely strips the whole thing back to give us a glittering gig in Regent's Park. Roman soldiers use their spears as mic stands, Mary Magdalene goes full Laura Marling and Judas Iscariot properly drops the mic as he dies. As for Declan Bennett's Jesus, he's like an angry Mumford; pouring his all into preaching through song, but privately exhausted by acclaim and adulation.

On a stage of rusted girders that suggest the ruins of the Twin Towers, a giant steel cross lies toppled on the floor. Ours, Sheader argues, is a world without faith and Jesus's followers, dressed in the baggy urban greys of All Saints, have something of the Occupy movement about them. They rave beneath colour dust and flares, railing at the old order – the cloaked Pharisees and marble-headed Romans – as they rally behind their Superstar. In their religious devotion, you see shades of millennial moral purity.

Tom Scutt gives the whole thing a glittery gloss. The Temple is a cloudburst of golden rain; Pilate wears silver laurels, the Pharisees, heavy gold chains – a marked contrast with those humble greys. Materialism finds its echo in fame; people stretching for some kind of permanence. Herod deifies himself in a bacofoil wrap, Judas plunges his hands into liquid silver and, en route to the cross, Jesus is blasted with gold-dust that sticks to his blood; the man and the icon in one, gold-gilt thorns digging into his flesh. The point, of course, is that even precious metals turn to rust.

What you see most of all, thoroughly pertinent today, is the mob – masses crowded behind figureheads and whipped into a frenzy. Drew McOnie's riveting choreography – often the most interesting thing onstage – rocks out with religious ecstasy at first, then turns righteous fury into trance. These bodies aren't in control of themselves; they shake and judder, lost in music. It's a far cry from the acoustic calm of Mary Magdalene. Whenever Anoushka Lucas steps in and sings, the air seems to lighten, the heat to lift.

And yet, and yet. Sheader's format snags between two stools. Handheld mics get in the way of acting; acting gets in the way of a genuine gig. Given the register of reality – actors eyeballing their audience, standing onstage and just singing – any characterisation at all starts to look concerted. Bennett's far-away glaze seems affected, his emotional explosions choreographed. David Thaxton plays Pilate like a Kaiser Chiefs tribute act, and only Tyrone Huntley, displaying vast vocal dexterity as Judas, cuts through the pretence. Despite Lee Curran's stadium-style lighting, which lifts the show as darkness descends, and Tom Deering's contemporarised orchestrations, it's still too polite for a real gig, too removed and too showy. In that, Sheader's gig becomes its own cross to bear.

Jesus Christ Superstar runs at Regent's Park Open Air Theatre until 27 August.