It seems like barely yesterday it was 2012 and the Rolling Stones were celebrating their 50th anniversary. But that’s how it is with this band. They’ve been around so long that time has ceased to be of consequence. Sure it would be great to be transported back to 1969, to Madison Square Garden and to their bluesy peak. Or to 1971 and Sticky Fingers. Or to 1976, when they were shambling their way through the desiccated funk of Hot Stuff.

But the newest song they play here – not counting their cover of Bob & Earl’s 1963 classic Harlem Shuffle – was Start Me Up, and that was released 38 years ago. It doesn’t matter: like old lions who can’t run down the antelope any more, they can still corner them against the rocks. Tonight they observe the first rule of stage business: so long as you still look nimble, there’s no penalty to age.

Keith Richards still seems to surprise himself with his enduring ability to riff his way through the band’s repertoire. He laughs as he heads on stage, and laughs as he heads off, and seems to be laughing most of the time in between. He is charming beyond measure, almost winking during his song to Anita Pallenberg: “Hey babe, what’s in your eyes? I saw them flashing like airplane lights,” on You’ve Got the Silver. Though on some solo numbers he looks like he might not make through – Before They Make Me Run, for instance, or his long, closing solo on Satisfaction – the audience carries him by sheer force of goodwill.

For Jagger, setting aside a recent heart-valve operation and his latest foray into fatherhood, he may finally be putting aside his role as the satyr of the late 20th century. His lasciviousness and grinding are less overt: he’s no longer running around out of breath, unable to enunciate. And he is connecting with the music as he’s rarely done, singing like he cares about the words. Jagger struts but never scampers, and when he plays the harp, as on their magnificent opus Midnight Rambler, he plays like no one else.

While Richards stabs his way through Sympathy for the Devil and Gimme Shelter, it’s the youngest member of the group – Ron Wood, 72 – who carries the show musically. Given the solo parts on Midnight Rambler and You Can’t Always Get What You Want, his playing is more defined than it used to be, or at least he’s taken more responsibility for it. Whoever is mixing the sound on this tour should get a medal: the guitars are kept high in the mix, so much so that in some sections when Wood plays his loud, dirty-sounding Gibson Firebird they seem to jolt the stadium itself.

The only worry here is Charlie Watts. He seemed barely present at the start of the show, but after the requisite acoustic centre-stage section he gets his swing together.

The Stones no longer seem imprisoned by their legend – indeed, the sex and drugs is now almost the least interesting thing about them. They’re playing better than they have done in years and seem grateful to be out, almost surprised at their good fortune. At last, the band are gracefully growing into the grandeur of their substantial achievement.

• This article was amended on 9 August 2019. The Sticky Fingers album was released in 1971, not 1973 as suggested in an earlier version.