Some of us, for the record, have always played the music. And some of us, also just to clarify, never stopped lighting the lights. That's because, for us in the cultural elite, we are always ready to meet the Muppets on The Muppet Show tonight.

When it was announced on Tuesday that US TV broadcaster NBC has commissioned a script for a new series of the Muppets, the reaction among critics, commentators and tweeters was, frankly, remarkable. It is rare that a four-decades old franchise can announce a return to TV and prompt such unabashed enthusiasm as well as a total lack of cynicism about quality control. Everyone loves the Muppets – that goes without saying. More surprising is how many people want them back, creating, satirising, karate chopping.

The Muppets are definitely having what Miss Piggy would call, with a proud tilt of her snout and a toss of her blonde mane, "un petit moment". The Muppets, the latest Muppet movie, opens in America this week and magazines across the nation have enthusiastically taken advantage of this to feature the cloth-covered puppets on their covers, in all their anarchic glory.

That film, though, has been slightly gazumped by the extraordinary documentary, Being Elmo, about the man behind possibly the most famous Muppet not on The Muppet Show but on Sesame Street. This beautiful movie has reminded audiences, if any needed reminding, that the Muppets were always more than just clever satirists but an integral part of American culture and society.

When Eddie Murphy dropped out of hosting the Oscars two weeks ago, a campaign was instantly launched for the Muppets to take his place. "Can Muppets Save the Day?" read the headline on the LA Times. "If the position were chosen by popular vote, the beloved Henson creations would likely come out on top," the journalist concluded. Sadly, the position is not chosen by popular vote and so the hosting duties went to another comedy throwback, Billy Crystal. But if the Muppets don't at least get to present an award, Miss Piggy should karate-chop Crystal. Hiii-yah!

Here is where I should, really, put the responsible disclaimer: my love for the Muppets is not without personal loyalties. My mother used to work for the Children's Television Workshop (CTW, now called Sesame Workshop), the non-profit organisation behind various Jim Henson shows. Incidentally, that is the coolest thing about me. It's all pith from hereon.

She worked on Sesame Street (The Muppet Show was, in fact, produced in England, as all English fans of the show tell me within one minute of first mentioning the programme – that explains the extraordinarily high number of cockney singalongs). Yet it's fair to say that I grew up in a household predisposed to watch all things Muppet-based. When I think back on my childhood, one of the first tableaux that comes to mind is me, my sister and our parents watching Sunday night screenings of reruns of The Muppet Show. It's a vision that seems so inspired by a 50s advert for TV sets that I'd doubt it – if my family didn't have a habit of making Muppet Show references to one another, if not on a daily basis then certainly on a weekly one, from Miss Piggy chasing her "Kermie" around, to the Mahna Mahna song. That Sunday night ritual, with my sister and me laughing at the slapstick gags, my parents laughing at the satirical ones, was as comforting as being tucked into bed later. It was like being told that everything, in the end, would turn out just fine.

The Muppets: a scene from the new film. Photograph: Scott Garfield

But in all professional and personal honesty, I cannot imagine that if my mother had worked on, say, The Magic Roundabout, I'd have loved The Muppet Show any less.

For those who have – bafflingly, tragically – not yet had the pleasure of seeing The Muppet Show, I shall explain. Presented as a chaotic variety show – replete with unruly audience, fond of throwing things at the MC, a little green frog called Kermit – with a different celebrity guest in every episode, The Muppet Show was Henson's attempt to break out of the children's-entertainer niche he found himself in after the success of Sesame Street. And to a certain degree, it was a success. Unlike Sesame Street, The Muppet Show was not interested in educating its viewers in anything other than funny pastiches of, say, Bohemian Rhapsody as performed by Gonzo and his chickens, or the brilliant comedy potential of pairing, say, Carol Channing with Miss Piggy for a rendition of Diamonds are a Girl's Best Friend. It was, at times, so surreal that one could only wonder how on earth it ever got on air, with scenes such as Miss Piggy trying to remove Rudolf Nureyev's towel in a sauna and Roger Moore beating the hell out of a bunch of Muppets. But it never sacrificed comedy for surrealism, let alone for the celebrity guest stars' egos. Like a Muppet-based version of Michael Frayn's classic play, Noises Off, or the great MGM musicals Singin' in the Rain and The Bandwagon, The Muppet Show was about the difficulties of putting on a show, with the main plots revolving around the relationships between the characters backstage (Miss Piggy pursuing a frazzled Kermit, Fozzie Bear always looking for and failing to find a joke), and the onstage set-pieces acting as comic relief. And as funny as those setpieces were, the personalities of the Muppets always struck me as the funniest part of the show, and seemed to amuse the celebrity guests most, too.

When I interviewed Dolly Parton for this paper over the summer, I tried to resist asking for my first question how she felt when she was honoured in Muppet form as "Polly Darton". I failed. Yet Parton, to her great credit, did not seem the least bit miffed: "Well, I was so excited! Who wouldn't be? I love that show. That was just a hoot! People your age still ask me about that – well, every week."

Stevie Wonder once said that playing Superstition on Sesame Street with the Muppets (and an amazingly enthusiastic small child in the background) was one of his career highlights. Roger Moore's Muppet Show appearances were absolutely his career highlight. Even Elton John – not a man known for handing out compliments lightly – has said that his 1977 appearance on The Muppet Show was "the most fun" he ever had and, I think it's safe to say, Elton was a man who knew fun from fun in the 70s.

Elton John performs Crocodile Rock on The Muppet Show. Photograph: TVTIMES/SCOPEFEATURES.COM

Children's TV shows – and despite Henson's attempt to escape the "children's entertainer" tag, The Muppet Show was, ultimately, for kids – always spark silly sentimentality. But the Muppets are not petrified in nostalgia. If anything, the original shows look better today than when first screened in 1976 (they ended in 1981). The genius of The Muppet Show was that it was ironic without being cynical, sharp without being cruel, sweet without being sappy, anarchic without being too chaotic, timely without being dated. These balancing acts sound impossible today, particularly if one spends too much time, as I do, reading internet blogs and pop culture magazines in which the only language spoken seems to be Snark. (Although it is perhaps inevitable that my favourite characters will forever be Statler and Waldorf, the snarky critics in the box, passing dismissive judgment on one and all. Well, what else would you expect?) It is a satire of a 70s variety show (especially through the house band, Dr Teeth and the Electric Mayhem) and of American TV of the time (Veterinarian's Hospital, the show's take on General Hospital, remains one of the best TV satires, ever) yet it is timeless for a simple reason: it is very funny and, crucially, very kind.

So, when I said the coolest thing about me was that my mother worked for the Children's Television Workshop, that was perhaps not entirely true. I do have another, possibly even more trumpity-trump card in my pocket.

In 1989, my family moved from New York to London. It would, our parents assured us, be just a temporary move. But within a year, it was obvious that we were never going to move back as a family and so, in March 1990, my mother, sister and I travelled back to New York to pack up our childhood home. I'd already had to give up my dog, my friends and my life (in that order of importance). Selling the apartment – my bedroom! – seemed the most unbearable change of all.

In an attempt to distract us from our prepubescent grief, my mother took us to Disney World afterwards. It was OK. On the third day, as we wandered from one hour-long queue to the next hour-long queue, I heard a familiar voice.

It just so happened that our trip to Disney World coincided with the filming of The Muppets at Walt Disney World, a made-for-TV movie in which the Muppets meet the Disney characters, and we were suddenly standing about 4ft away from Jim Henson himself, bearded, sun hatted and in a lavishly patterned shirt, giving the frog hoiked up on his arm that reassuringly familiar voice as well as that endearing personality. Behind Henson, Frank Oz was adjusting Miss Piggy, doing her splendid diva head-waggle as she addressed her "Kermie". During a break, Henson smiled and said hello. Kermit shook my hand and hoped I was enjoying my vacation more than he was. He didn't like the heat that much; he wanted to get back to the swamp. I had met the Muppets, even if not, strictly speaking, on The Muppet Show tonight, and it was easily the most magical moment, quite possibly, ever.

Two months later, Henson died suddenly at the cruelly young age of 53.

We went back to London where we lived for the next 20 years and made new lives. When The Muppet Christmas Carol came out in 1992, the first Muppet movie since Henson's death, my family and I all went to the cinema together and we were relieved, we agreed, to see that it was up to the high Muppet standard. No matter what details changed and how life moved on, there will always be the Muppets. And everything, in the end, turned out just fine.

• The Muppets film is released in the UK in February 2012