Charles Darwent on Paper at the Saatchi Gallery: He's all over the papers again

A free show of throwaway material may sound lightweight, but it's brighter and breezier than the heavy-handed hype

Ashow called Paper. Lighter to put on than the Royal Academy's Bronze, no doubt, but potentially rather larger. Exclude canvas and panels of various kinds and what two-dimensional art isn't on paper? The Saatchi Gallery is big, but not that big.

It is the Saatchi Gallery, though, which means that its interests are limited. Where Bronze took in 5,000 years of art-making, Paper takes in around five (2008-13). It also doggedly excludes the kind of object suggested by the words "works on paper". This is Saatchi-art, which is to say hip and new. So Paper was bound to be an interesting show, although it ends up being less interesting than it might.

The problem, as often, is with the hype. It's fair to say that, outside the realms of sculpture, bronze doesn't have much resonance for most of us. Paper does. Consider the thing you're reading, the thing it's printed on and the chilling words "paperless society". Paper has a moral significance, exists at a critical moment. The Saatchi show promises art that addresses these issues, that is not just work-on-paper but work-of-paper. For much of the time, though, it doesn't deliver.

That doesn't mean that some of the objects in Paper aren't good. A few even fit the paper-centric remit promised by the show's publicity handout (printed, of course, on paper). So let us start with the positive.

The Colombian artist Miler Lagos – at 40, antique by Saatchi standards – turns the clock back by compressing waste paper into cylinders and carving these into branches. From wood were ye made, and to wood shall ye return again: it is the ultimate in recycling. Much of Colombia lies in the Amazon basin, where trees are being cut down very much faster than Lagos can resurrect them – a sobering truth soberly spelt out by his floor installation.

There is something of the same spirit in Yuken Teruya's LVMH series, paper bags from posh shops – Pucci, Christian Dior, Givenchy – which Teruya has fixed to the gallery wall in a row, making a sequence of diminutive box spaces. In each space is a single, Godot-ish tree, fine-cut from the paper of the bag itself and pushed up into standing position.

It is all very Japanese, which is part of the point. Cut paper has a specific history in the art of Japan, so that the idea of a paperless world encompasses the death of a national tradition. At the same time, Teruya reminds us about the trees cut down to feed this tradition, the status of art as a luxury good. Each of his bags is a miracle of complexity, a moral drama played out in miniature. They would be worth going to see even if you had to pay to get into the Saatchi Gallery, which you do not.

So far, so good. Next down in terms of see-ability is work that is interesting but for which being made of paper seems pretty much coincidental. Nina Katchadourian's Lavatory Self-Portraits in the Flemish Style are shot mid-flight in the toilets of passenger planes. (One imagines queues of fellow passengers cursing her name, but let that rest.) Nipping in with her mobile phone camera and raiding the kind of things you find in airline lavs (disposable seat covers, tissues, etc), Katchadourian photographs herself done up like a 15th-century Flemish woman.

Like the digital images she takes, the materials Katchadourian uses are throwaway. There is something disturbing about the clash of traditional ideas of portraiture – the point of Van Eyck and the rest was to immortalise, after all – and the modern cult of the disposable. These are astonishing images, and many of Katchadourian's props are made of paper. But is her work really about paper?

Contrariwise, Marcelo Jacome's Planos-pipas no 17 could be made of nothing but, consisting as it does of Brazilian paper kites. Its flying-dragon form celebrates the material's lightness. Jacome's lovely, room-long work has the feel of a Futurist drawing trying to capture motion, of kite-swoops frozen in space and time. But it doesn't address the moral and historical problems of paper, paper's here and now.

Many works in the show are intriguing, but few have very much to do with the material of which they are made. They are objects such as photographs and watercolours, for which paper is merely a useful vehicle. But I don't mean to cavil, and I wouldn't, had Paper not oversold itself, or been oversold.

To 29 September

Critic's Chopice

Seizure started life in an old London council flat: Roger Hiorns coated the interior with copper sulphate, prompting sharp, rich blue crystals to form all over it. It's just been installed in a new environment, at the Yorkshire Sculpture Park. For very different site-specific art, visit Uncommon Ground, which brings together the work of British land artists 1966-1979. Get earthy with Antony Gormley (pictured), Susan Hiller and others at Southampton City Gallery (to 3 Aug).