BTS arrive for their first ever UK shows by private jet. They have been using it on the US leg of their world tour, which culminated in a show to 40,000 people at New York’s Citi Field on 6 October, three days before playing to as many people again across two nights at the O2 Arena in London. They have racked up two US No 1 albums and billions of global streams, and were recently invited to the UN as Unicef ambassadors, where their charismatic leader, RM, made a speech, in English, on self-acceptance. Milestones such as these are monumental for any artist, but in reaching them BTS – rappers Suga, RM and J-hope and vocalists Jimin, V, Jin and Jungkook – have changed the face of pop, as the first Korean group to reach the upper echelons of the western music industry.

Ethereal-looking Jimin broke down at the end of the Citi Field show. The band have played similar-sized shows in other countries, but the US has always been the final frontier for K-pop – a market that has been attempted many times with only minor successes by acts such as Big Bang, EXO, and 2NE1’s CL. “We feel it all the time,” says Jimin. “On this tour we played some very large venues, and it makes us see that people really love us. Being inundated by all these emotions, it kind of got to me.”

In a hotel in London, ahead of the UK shows, security stake out the hallways. Burly men accompany band members to the toilet. BTS have reached that dissociative level of stardom where they are handled like china dolls. “We know that popularity is not for ever,” RM says with a smile. “So we enjoy the ride, the rollercoaster, and when it ends, it just finishes. We’re on the jets and in the stadiums, but I don’t feel like it’s mine. It’s like we just borrowed it from somebody.”

BTS are the brainchild of veteran writer and producer Bang Shi Hyuk, who formerly worked at the K-pop entertainment giant JYP, then formed Big Hit Entertainment and debuted BTS in 2013. The normal practice of K-pop is to oversee every element of the life of young “idols”, as they are known in Korea. However, Bang gave BTS autonomy to run their own Twitter and vlog from their studio, and for the rappers to write alongside Big Hit’s in-house production team. Their lyrics are emotionally vulnerable and socially conscious, sometimes bordering on angry, and go against K-pop’s grain: Baepsae, which translates as “silver spoon”, defends their “cursed” generation.

Critics have tried to unravel the secret of their US success: many credit social media with spreading their message, but BTS’s fans, known as Army, flag the music and lyrics as the reason they have connected so deeply. It’s this, plus the end of One Direction, the growing interest in K-pop in the US, and BTS’s endless stream of visual content (from behind-the-scenes footage to reality shows) that reel in the curious and hook them with the force of the group’s personalities. In time-honoured boyband fashion, they offer something for everyone.

Like all pop stars with gigantic, powerful fan bases, BTS tread a delicate line between celebrating their admirers and potentially alienating them. “Fame is like a shadow,” says Suga, their most serious member. “There’s light and there’s darkness; it’s something that follows you constantly and not something you can run away from. But people tend to respect our privacy. We go to art galleries a lot and people don’t really bother us, then after we leave they’ll make a [social media] post.”

“If it gets too much and it crosses a line, then it can be a source of stress but for me, at least, it’s a sign of their love,” says J-hope, a former street dancer. On a recent album cut, Pied Piper, they playfully admonished the obsessives: “Stop watching and start studying for your exams, your parents and boss hate me … You already have plenty of my pictures in your room.”

That surprising honesty – in K-pop terms – underpinned the concept of their recent Love Yourself album trilogy (Her, Tear and Answer), which charted a narrative around, unsurprisingly, learning to love oneself. RM’s speech at the UN echoed this theme: “No matter who you are, where you’re from, your skin colour, your gender identity, just speak yourself.” This relatively anodyne statement resonated in South Korea, where the president publicly opposes homosexuality.

During their career the band have used Haruki Murakami, Ursula K Le Guin, Jung, Orwell, Hesse and Nietzsche as inspiration. The latter figures notably in the theory of fate that is woven through Her, whereby love is destined and must therefore be unshakable (only for it to fall apart on Tear). As 80s indie fans did, BTS’s Army now read these writers in order to fully understand the band’s vision, while spending serious money on Bluetooth-programmed light-up sticks for their concerts.

(From left) V, Suga, Jin, Jungkook, RM, Jimin, J-hope. Photograph: Young Ho/SIPA USA/PA Images

For many, however, BTS symbolise an industry that is little more than a high-functioning bubblegum machine. K-pop is perceived as cruel for its intensive training system, which can start when artists are seven years old and last for 10 years with no guarantee of a group debut; and for its harsh approach to idols who struggle with exhaustion and their mental health. Many have fainted onstage, while Super Junior’s Leeteuk quietly set up a now-defunct peer group, Milk Club, for idols dealing with depression. Meanwhile, fans are portrayed as mindless teenage girls. “It’s pointless to argue or fight about it,” Suga says, gruffly. “Frankly, I can’t understand people who want to put down a certain type of music, whatever that might be. Classical music was pop music in its own age. It’s a matter of taste and understanding – there’s no good or bad, there’s no highbrow or lowbrow.”

BTS’s music began as old-school R&B and hip-hop, but has since incorporated a myriad of genres, from EDM to South African house. The lyrics, too, have become increasingly complex, closer to prose than simple moon-June-soon pop. In many respects, BTS fit the mould of a classic boyband – they look and sound great – but they are also grown men who cry, embrace and expose their vulnerabilities and failings even as a culture of toxic masculinity thrives on- and offline. It strengthens their messages of strength, love, hope and acceptance beyond what boybands have offered before.

K-pop idols work intensively, in a world where a few careers will last more than 10 years, but many are over in just 12 months. This year BTS have released three albums (two Korean and one Japanese), toured the world and produced a third series of their travel reality show, Bon Voyage. Their schedule is planned down to the minute. “I think there were times we were pretty close to burning out,” admits Suga, “but it’s inevitable and it’s the same for people in any profession.”

Current and former idols have shifted towards acting, appearing on South Korea’s variety TV shows, and explored solo careers. Suga’s interests include architecture and lighting. Jungkook, the youngest member at 21, has taken up documentary-style film-making, his most recent short capturing the extremes of his life – the intensity of the stage and the stillness of the aftermath. He says he feels “a lot of happiness when I think about things I can do in the future”. He has energy to burn – he will later give himself a minor heel injury before the first London show and spend it confined to a stool, tearfully apologising for not fully participating.

During a recent live session on streaming platform VLive, V, whose slightly hoarse voice gives the group a soulful edge, played snippets of solo work to much buzz. BTS’s rappers have already released self-written and produced solo mixtapes, but the vocalists have yet to follow in their footsteps. “I’m working on it,” offers Jungkook, when J-hope begins laughing.

RM weighs in, amused, “He’s getting ready for too many things! Films, boxing – he’s planning so much that no one knows when it’s coming out.”

A good-natured squabble breaks out. “When J-hope gives me the beats, maybe I can get started on my tape,” deadpans Jin, the oldest member at 25.

J-hope feigns indignation. “I gave him beats! He liked what I gave him!” he says as Jin cackles at the ceiling.

“On all the songs I make,” V chips in, having sat back for most of the interview, “I feel there’s something that’s just not there. I have a shortcoming, I can’t finish a song, I need someone to help me. When I come up with something I can put out, I will.”

Suga jabs back. “It’s going to be about 20 years then.”

For their fans it’s this kind of playful teasing and natural camaraderie that makes BTS so appealing. For the band, their connection helps support their frenetic work pace.

Openly ambitious, Suga has repeatedly stated that a Grammy win is his next goal and recently added playing the Super Bowl halftime show (71,000 people in the arena; 120m watching at home) to the list. Either could be the thing that cements BTS’s status as household names. Right now, neither seems unreachable. “We want to show as much as we can,” says Jimin, his gaze unwavering. “We only want to be able to show our best.”