Gustav Klimt: Painting, Design and Modern Life in Vienna 1900, Tate Liverpool

A moral conundrum this week: is it possible to know too much about an artist? Can more – in the sense of context and background – ever be less? Is honesty always the best policy? If Tate Liverpool's new Klimt show is anything to go by, then the answers to these questions are yes, yes and no, in that order. Billed, rightly, as "the first comprehensive exhibition of Gustav Klimt's work ever staged in the UK", this seems likely to disappoint precisely on the grounds of its comprehensiveness. For the Tate's show, in doing what it says it will, both sidelines Klimt and reveals him to have been two things forbidden to the Modernist canon: a decorator and a craftsman.

Although the subject of this exhibition is one of the world's most recognisable painters, this is largely on the grounds of his work's prettiness. For the past 40 years, no undergraduate bedsit has been complete without its Klimt poster – preferably something from the artist's later career, one of the so-called Malmosaike (painted mosaic) pictures, such as Judith II (Salome).

When Estée Lauder's son snapped up the 1907 Portrait of Adele Block-Bauer two years ago for $135m (£68m), the work's price was a multiplier of its prettiness. It was the goldest Klimt around, therefore the best.

As a result of this, the Austrian has tended to suffer from what you might call the Athena print effect: because his work is hung, in reproduction, on a million teenage walls, Klimt has become the kind of artist who painted teenage-wall pictures. This ex post facto reasoning seems especially unfair to a man who was, in various ways, extraordinarily cerebral and radical. So one aim of a good Klimt show might be to explain to a poster-buying public exactly why the image over its bed is something more than pretty.

Perhaps the best-known aspect of Klimt's radicalism is the part he played in the Vienna Secession, that salon des refusés set up in 1897 to defy the stultifying conventions of late-Hapsburg Austria. A central aim of the Secessionists was the unification of art and design – everything from painting to couture, all brought together in a vast Gesamtkunstwerk with its own workshop, the Wiener Werkstätte. In theory and in practice, the Secession echoed the British Arts and Crafts movement. The Secession building looks oddly like a Charles Rennie Mackintosh for the good reason that it was heavily influenced by him. One of the Werkstätte's founders and Klimt's patrons, the financier Fritz Wärndorfer, was a friend of Mackintosh's, and introduced his work to Vienna.

All of which is to say that the Secession, a supposed touchstone of Modernism, also had its anti-Modernist tendencies. Its Ruskinian mistrust of manufacturing and emphasis on beautifying the world was already beginning to look old hat by the time Klimt painted Judith II (Salome) in 1909. And Klimt, too, was in many ways a retardataire artist, much of the work in this show looking dated for its time. Two Girls with an Oleander, finished in 1892, could happily hang in the Lady Lever Art Gallery next to a Millais painted 40 years earlier. His Portrait of Hermine Gallia (1903-4) is nothing short of Whistlerian. It is only in the last decade of his life that Klimt really became a revolutionary, and then not for long.

The trouble with this scrupulous, well-intended exhibition is that it doesn't give us enough proof of that late avant-garde flowering. Broadly, the show is organised by project and patron rather than by Klimt's artistic development: thus we see Portrait of Marie Henneberg in the context of the Villa Henneberg, alongside architectural designs by Josef Hoffmann and photographs of the overmantle into which Frau Henneberg's square-format portrait was made to slot. With Klimt's Portrait of Eugenia Primavesi come the contents of the Primavesis' sideboard – pen-wipers, fruit-baskets and coal scuttles– all designed by Klimt's friend, Koloman Moser.

These things are certainly stylish, however, it is difficult to draw an aesthetic line between Moser's silverware and the handsome Jugendstil frame of Frau Primavesi's portrait, and between that frame and the picture it surrounds. And this is just as it should be, the point of the Secession's Gesamtkunstwerk being to erase division and hierarchy.

But while all this may help us to understand Klimt, does it really help us admire him? The problem with this show is, as I say, that it contains only about 20 of his paintings, and most of these are, in the literal sense, decorative. Only Klimt's wonderful erotic drawings and a handful of late canvases show what he might have become had he not died of a stroke in 1918 at the age of 55.

I don't know what happened to Klimt around 1905, but his work suddenly stopped looking backwards and began to look forwards. Perhaps he read Three Essays on the Theory of Sexuality, published the year before by that genuine giant of Viennese Modernism, Sigmund Freud. At any rate, he began to paint as Arthur Schnitzler wrote, with rawness and clinical honesty. The veined hands of the left-most figure in Three Ages of Woman lead directly to Egon Schiele, and on from him to the German Expressionists (maybe even to Freud's grandson, Lucian). I very much doubt that, if Klimt had made it to 70, we would find the work of his last 15 years reproduced as posters for bedsit walls. But that, unfortunately, is where this show leaves him, even if it does so with the best will in the world.