'Zweimal Hitler bitte," I requested at the ticket desk for the Hitler exhibition at the German Historical Museum, meaning "two tickets please" but saying literally (and, I confess, as a little experiment) "two times Hitler please". The middle-aged lady on the desk neither batted an eyelid nor missed a beat. "Den gab's aber nur einmal," she replied, in the characteristic Berlin accent: "but he only existed once" or "there was only one of him".

Quite right too. And Gott sei dank. For decades, probably centuries to come, the name of Hitler will remain a worldwide synonym for evil. In a secularised Europe, he is a more frequently encountered personification of evil than the devil. In a Californian swimming pool this summer, I saw an American dad offer himself as the "bad guy" to be shot at by the kids with water-pistols. "Hitler!" they shrieked, as they squirted him with water, "Hitler!"

But there is no justification for viewing the problems of today's Germany, and Europe's occasional difficulties with it, through the prism of Hitler. This is not just a matter of the 65 years that have elapsed since his death. Rather, it is a measure of Germany's own achievement in remaking itself as a liberal, democratic state and open society. One of the ways it has done that is by repeatedly facing up to its own difficult past – most recently in a historical study, commissioned by Joschka Fischer when he was foreign minister, showing just how deeply the gentlemen of the German foreign office were implicated in the Nazi regime. So having an exhibition about Hitler is evidence of how far Germany has come from Hitler.

This is not, it must be said, a great exhibition. Though full of interesting detail, it is rather cluttered and nervous. It keeps feeling the need to remind you that Hitler was a Bad Thing. It doesn't quite dare to draw you into its ostensible theme – why Hitler fascinated and enthused so many Germans. It would have been good, for example, to have a darkened room where the visitor could be exposed to the full force of that fascination through the cinematic eye of Leni Riefenstahl. But everything about it is perfectly sober, and so are the hushed visitors who cram its rather tight spaces.

In one wall-mounted video loop, it rather cleverly pairs the famous scene from Charlie Chaplin's Great Dictator – showing Napaloni, dictator of Bacteria, arriving by train to be greeted by Adenoid Hynkel, dictator of Tomania – with some actual newsreel footage of Mussolini arriving by train to visit Hitler in 1937. (The real leaders do visibly compete in strutting and arm-waving.) Now that Great Dictator scene is, of course, very funny; but my wife and I seemed to be the only people laughing. No laughter please, we're German – and, more to the point, Germans visiting an exhibition about Hitler.

Yes, if you hunt through the guest book at the end you can find one silly graffito, in a childish hand, calling Hitler "cool". But most of the entries, in many languages, are sensibly appreciative of what the exhibition is trying to do.

At one point only did I feel a shudder-inducing connection to current German debates. On display was a Nazi poster showing how people of "inferior race" could overtake healthy Aryans as a proportion of the population, because of their higher birth rates. I have just been reading a hugely controversial book called Germany Abolishes Itself, by a Social Democrat and former Bundesbank director, Thilo Sarrazin. Among a number of perfectly sensible arguments about the insufficient integration of immigrants and the burdens of the welfare state, Sarrazin makes the (stupid) case that Germany is getting more stupid because it has taken in so many uncultured Muslims. I am not for a moment wanting to imply that Sarrazin is some kind of a closet Nazi, but you would think that a German author might display a special sensitivity when it comes to claims about genetic characteristics of ethnic groups.

These side-echoes aside, however, what has come to be called the "Sarrazin debate" is really not that different from the controversies about Muslim immigration in Holland, Spain, Italy or Britain. The German debate is not worse, though, alas, also not better. In this respect, as in others, Germany has become a "normal" European country.

As for soldiers charging around in tanks, the only Europeans who do that seriously today are the British and the French – and even they can only do it, as they have just boldly acknowledged, by sharing their resources. Like most other European armies, the Germany army does many valuable things, but fighting is generally not among them. The Bundeswehr is closer in spirit to the Salvation Army than it is to Hitler's Wehrmacht.

What today's Germans do instead, with fiendish ingenuity, discipline and efficiency, is to make things that people in other countries want to buy. We may envy them, but who can blame them? Absorbing the gobsmacking bill for German unification (about €1.6 trillion), consensually managing down unit labour costs (at a time when these were soaring in countries like Greece), exploiting the advantages given it by the euro (a stable world currency, eurozone neighbours who can't compete by devaluation), seizing new market chances in China and elsewhere – the German economy thrives while others falter.

Its success does, to be sure, rest on a paradox: if everyone else behaved like the Germans (both exporting and saving more), as the Germans say they would like their partners in the eurozone to do, then the Germans themselves could not go on behaving like the Germans. Their export model depends on extravagant others buying their goods.

In Europe and the wider world, Germany is increasingly inclined to pursue its own national interests, on its own account if need be (for example, in its bilateral energy relationship with Russia), and to respond defensively to domestic pressures: whether it be slowing down visa-free travel from the Balkans, to placate a distinctly Sarrazinesque German public opinion; or seeking eurozone-related changes to the Lisbon treaty, not least to fend off its own eurosceptic constitutional court. The British and the French, who have always pursued their own national interests, are the last who have a right to complain.

That said, we do sorely miss the exceptional German engagement in Europe, which was such a salient feature of the federal republic's foreign policy from Konrad Adenauer to Helmut Kohl. The European project is stalled not least because the German motor is no longer driving it forward. It is much clearer today what Germany wants from Europe than what it wants for Europe. The German foreign minister, Guido Westerwelle, tried to elucidate this in a recent speech here in Berlin, but the answer somehow got lost in a quivering blancmange of neo-Genscherite waffle.

The truth is that Germany still needs Europe, as Europe needs Germany: not for the old reasons, which had to do with Hitler and the world of 1945, but for new ones, which have much more to do with Hu Jintao and the likely world of 2045.