Sometimes it’s easy to forget just how big artists have become until you watch a few thousand people try to rush their stage.

A sold-out crowd flocked to see Tame Impala play Melbourne’s Sidney Myer Music Bowl on Saturday night, their first Melbourne headline engagement in the two years they’ve spent criss-crossing the globe. That means 12,000 or so people, and that’s not counting the hopeful picnickers craning their necks from beyond the fences and perched in the branches of the venerable trees surrounding the venue. When Kevin Parker and his touring band began to unfold the mysteries of Let It Happen, most of the stalls emptied and surged to the front of the stage.

Parker, whose approach to the writing, recording and production of Tame Impala records recalls the megalomaniacal commitment of Boston’s Tom Scholz (though Parker is clearly much nicer), is accompanied around the world’s stadia by a four-piece band: Jay Watson on synths, Dominic Simper on guitar, Cam Avery on bass guitar, and Julien Barbagallo on drums. They’re a formidable live presence, accompanied by a charmingly old-fashioned light show whose psychedelic projections wash Parker and co in blobs of colour and movement.

“Holy shit, you go back so far,” Parker exclaimed, eyes straining to the back of the crowd on a distant hill during a break in the set. “You guys wanna know another fun fact? This is the biggest Tame Impala show there’s ever been. All 12,000 of you.”

Whether or not that’s an apocryphal statistic – Tame Impala are, after all, a very popular band – the crowd reacted to this announcement with apoplexies of joy.

And the alternate joyfulness and wistfulness of Saturday’s set is what makes certain criticisms of Tame Impala seem unduly mean-spirited (especially as such criticism tends to be a levelled by the tiresome cowboy-booted members of the indie rock scene, whose favourite artists have been reheating the same Rolling Stones leftovers for decades). Only the dense would deny that Tame Impala more than occasionally recall, among other things, late-period Beatles, the Grateful Dead, the Moody Blues’ In Search of the Lost Chord, but the jackdaw-ish Parker churns these touchstones into something appealingly new.

The band tore into material from 2015’s Currents before dipping deeper into the Tame Impala catalogue, but it was the new record’s soaring, mournful songs that seemed to most bewitch the crowd: break-up anthem Eventually seemed to give the crowd a similar catharsis to the one I imagine Abba’s Knowing Me, Knowing You did at the same venue in 1977.

As dreamily engaging as the new material from Currents is, it’s the heavier moments (perhaps a relative concept) from 2012’s Lonerism that stand out live: the stoner churn of Elephant expands into a shuddering behemoth live, so exciting it was almost a disappointment to return to laidback psychedelia afterwards. Similarly, there’s a heft to Mind Mischief’s fuzzed-out guitars and Abbey Road-esque drums that expands to fill the live space. Both songs were standouts.

Is Parker too tender a soul to fully commit to the darker fringes of psychedelia? The whorls of weed smoke that clung to the audience and mingled with the smoke machines on Saturday night suggest a fanbase who’d be happy to be propelled into a dark night of the sonic soul and beaten around the head by some Sabbath-isms.

There’s something about Parker’s wistful psychedelia that – though tonally quite different – recalls the gentle, heartfelt music of the 70s west-coast boom and its accompanying raw emotion, something notably missing from the beige singer/songwriter fare of recent years. He’s not afraid to open a vein on record, and does so in an appealingly earnest, almost teenaged manner (“They say people never change, but that’s bullshit” a representative lyric).

Towards the end of Saturday’s show, Parker announced, “We’re going to play a song we’ve never played [live] before.” As he and his touring band launched into the aching Yes I’m Changing, fireworks bloomed over the hill and lit the ecstatic faces of 12,000 dreamers, who instantly forgot the evening chill – and their homework, bills and responsibilities – and were transported to a gentler dimension.

• Tame Impala are touring Australia until 21 November and then continue their world tour in New Zealand