Over the past few years, the National have become the most important band in modern rock music. The strange thing is that this has happened at a point when their own work has perhaps lost a little of its earlier intensity. They’ve become important because they have come to represent something to other artists: a kind of adventurous but accessible integrity. The brothers who are the musical core of the band – guitarists Aaron and Bryce Dessner – have been so in demand that they have worked with, between them, (deep breath) Taylor Swift, Ed Sheeran, Michael Stipe, Sharon Van Etten, Bon Iver, Philip Glass, Steve Reich, Paul Simon, Sufjan Stevens, Nico Muhly, Jonny Greenwood, Bruce Springsteen and Ryuichi Sakamato. And those are just the ones you might have heard of. Then there are the three festivals that Aaron helps run. Their tentacles are everywhere.

It’s easy to see why everyone fancies a piece of them. At Ally Pally – hands down the worst major concert venue in London: terrible sightlines, awful sound – they showed their gift for music that manages to be as accessible and unspecifically emotional as Coldplay, without ever feeling as though they are scrambling around for the lowest common denominator. The band – the Dessners, plus singer Matt Berninger and sibling rhythm section Bryan and Scott Devendorf, supplemented by a couple of horn players – showed their mastery of tone and texture, and on the set-closing trio of ‘Pink Rabbits’, ‘England’ and ‘Fake Empire’ they were breathtakingly good.

But I’ve never been able to buy into them completely. I’ve seen them a lot over the years, and like a group I truly love, Yo La Tengo, I never know which way I might turn when I see them live: will I be bored rigid, yearning for songs that actually go somewhere rather than churning intensely on the spot, or will I be hypnotised by the mood and luxuriant sound of it all? At Ally Pally, I was halfway between the two: they played for just shy of two-and-a-half hours, which is an awfully long time for a band short on copper-bottomed, undeniable bangers. At 90 minutes I’d have been transported. At an hour more, I was wondering when I might get home. My feeling is that the best National-adjacent record is Taylor Swift’s Folklore because you can be pretty sure she wanted definite purpose to all the gauzy beauty.

Then there’s Berninger’s collection of tics, which veers a little close to self-parody, and his odd, distracting vocalisation (he seems to swallow the ends of words which means he rarely seems to be singing complete sentences). The lyrics, too, were poised in a strange place between profound and meaningless: ‘Bloodbuzz Ohio’, one of the very few undeniable bangers of the night (with a perfect, mysterious title), has a chorus that runs: ‘I was carried to Ohio in a swarm of bees/ I never married, but Ohio don’t remember me.’ It must mean something, right?

Still, when I looked at Instagram the next morning, it seemed as though every single person I knew had been at the show, and everyone thought it was one of the greatest things ever. So I’m perfectly prepared to concede that this is about my shortcomings rather than the National’s.

Watching PiL at the Forum in Kentish Town, the thought occurred: oh, this is just like going to see the Who. Not that PiL sound like the Who: more that John Lydon, like Pete Townshend, is trapped in a furious persona he created decades ago. He has to wheel it out for the delectation of beered-up geezers who don’t seem terribly interested in the vulnerability and pain and terror at the heart of the best songs, but want instead to throw their plastic pint pots and push each other around.

Lydon keeps on working – there was a pretty decent new PiL album this year. The best songs, though, were the ones from their second album, Metal Box, which still sound like a dream turning into a nightmare in real time. And the most moving moment came after the music had stopped, when Lydon – freed at last from having to tell people to eff off – asked the crowd to shout an ‘Aloha!’ for his late wife Nora. When they did, he screwed up his eyes and his shoulders shuddered. And he didn’t look like a caricature, furious, or confrontational. He looked human and weak and entirely unlike the Lydon he usually offers to the world.