Why did the EU ever look like a good idea?

The EU was set up in an attempt to solve serious problems which faced the European countries in the 1950s to 1970s. Those problems have now entirely disappeared, for unrelated reasons, but the Union they spawned hangs on as a curious relic.

Problem 1: Protectionism

After the first world war, trade between nations collapsed under protectionist policies, bringing in the Great Depression. After the Second World War, there was a fear that the same could happen. The European Community at least provided for a degree of freedom of trade within Europe, even if more distant markets were closed.
The level of restraint of international trade that was normal as late as the 1970s would be unthinkable in any major economy today. The EU has not kept pace, and by now it increases protectionism rather than decreases it.

Problem 2: Communism

Early in the cold war, there seemed a strong danger that one or more of the Western European countries could elect communist parties to power and join the Soviet bloc. By locking the economies of these countries closer together, the EU would make it more difficult for left-wing governments to change sides in the cold war.
This reason obviously no longer applies.

Problem 3: Imperial Rivalry

This was not really a current problem by the 1960s, but it was at least a recent problem. Through the 19th Century, the major European powers had gained or sought to gain overseas colonies. This resulted in wars fought over control of colonies, and intense rivalry over access to or control over trade routes. Each power feared that its rivals would be able to cut it off from access to raw materials or markets. This culminated in the First World War.
By now, overseas colonies of European countries have virtually all gained independence, and the traces that remain are seen as economic liabilities rather than assets. The world trade system based on the WTO guarantees all western nations access to global markets, and while protectionism lives on it is mainly driven by special interests within countries rather than strategic or military advantage.