Georges Seurat had kaleidoscope eyes. He saw in limitless colours, that swarm and bubble on his canvases in galaxies of tiny dots. Choosing random, barren subjects – an empty harbour, a rock – he found endless wonder in the most banal reality. In his 1888 painting Port-en-Bessin, a Sunday, myriad blues and whites create a hazy sky and mirroring water while a railing in the foreground explodes into purple, brown and orange as if it had a lurid spotty disease. Seurat only lived to the age of 31, but he inspired an entire art movement, the neo-impressionists, who copied his “pointillist” method.

Yet in a coarse-grained approach to this fine-grained art style, the National Gallery struggles to tell a different story. The neo-impressionists didn’t just paint dots, they dreamed of revolution. And by the way we shouldn’t call them by the evocative nickname “pointillist” because they didn’t like it.

You can understand why the National Gallery might long to shake our perceptions of late 19th-century art. This was a time of astonishing experimentation, especially in Seurat’s birthplace Paris, when artists were rethinking how we see, making it easy for Matisse and Picasso to take the final step to modernism in the 1900s. Yet, who doesn’t find impressionism and its aftermath soothing, lovely, fundamentally happy? Here we are invited to see the iron fist under the velvet evening light, bite the grenade in the chocolate box. As a result, the sublime is often turned into the ridiculous.

New picnic … study for In the Time of Harmony by Paul Signac. Photograph: Paul Signac (1863-1935)/© Collection Kröller-Müller Museum, Otterlo, The Netherlands. Photographer: Rik Klein Gotink

A drawing by Paul Signac, Seurat’s first disciple, shows what the caption calls “ordinary people” enjoying “rest, social harmony and the bounties of nature”. A man reaches up for some fruit while a woman dangles a cherry or grape for a child. I think the word we’re looking for here is “picnic”. I’m not doubting Signac’s radical politics when he did this design for a mural called In the Time of Harmony, but they are not present, artistically. Signac lived a lot longer than Seurat and by 1904 was, like any good anarchist, residing in Saint Tropez. Henri Matisse stayed with him there and that year, under Signac’s influence, painted the last pointillist masterpiece, his manifesto for pure pleasure Luxe, Calme et Volupté. I wish it were here.

Having failed to prove the neo-impressionists were on the verge of toppling the Third Republic with dots, the exhibition moves on to claim they challenged the idea that “a portrait should capture a specific likeness”. No, they didn’t. Most of the portraits here are highly conventional under a thin pointillist veneer.

This is the trouble when you base an exhibition on a single collection. Most of this exhibition comes from the Kröller-Müller Museum in the Netherlands, founded by the early 20th-century art collector Helene Kröller-Müller. She was a serious, perhaps melancholic north European whose true love was Van Gogh: her museum has more than 90 paintings by him. Kröller-Müller sees the pointillists through a northern lens, with a late Romantic mind moved by spirituality and introspection. Combine this with po-faced curating and the fun of this art movement is almost snuffed out.

Utterly fantastical … Le Chahut by Georges Seurat. Photograph: Photography Tom Haartsen/Collection Kröller-Müller Museum, Otterlo, the Netherlands

In 19th-century France it was pleasure, not inner pain, that was valued. The modern world looked bright, bold, joyous. And here it is. At the heart of this show, Seurat blows away his imitators with his mind-blowing vision, the jewel of Kröller-Müller’s neo-impressionist paintings, his realist yet utterly fantastical 1889-90 painting Le Chahut.

A row of dancers, side on, are performing a wild yet disciplined can-can, legs high in the sky in a regimented line, while the orchestra play below them and a caricatured man with a face like a pork chop looks up the women’s skirts. The dancers are caricatured too, the self-delighted women and the men who alternate with them in the line. It’s a painting you are meant to smile at and be dazzled by. Seurat’s colours confound. The back of a bass player is a pulsing shimmer of purples, the red stockings of the dancers have a similar pointillist shimmer, the wall of the cabaret theatre vibrates with gold and blue dots.

The difference between Seurat and his imitators is that his vision is totally encompassing: there are no forms in his art that are not completely defined by tiny points of light. That means every bit of skin, skirt and slobbering voyeurism is bathed in a dreamlike lurid greasepaint glow. It is, in every sense, dotty.

Seurat sees something comical yet seriously modern in this. The dancers move like well-oiled machines: they seem unconscious, the moves are so ritualistic and ingrained. The musicians too are on automatic, lost in the chopping craziness of the music. Seurat here anticipates Duchamp and Picabia who would equate sex and machinery at the start of the 20th century. He is ahead of them all, exposing the fabricated nature of the “reality” we think we see – and giving that a social dimension when he paints people as mechanical dolls.

Paris is a great machine, beautiful but heartless. Oh, wait. That actually is pretty radical.

Radical Harmony: Helene Kröller-Müller’s Neo-Impressionists is at the National Gallery, London, 13 September to 8 February