> The purpose of is the EU single market
That's also a misconception. The EU single market is an important project, but there is a constant failure to understand the actual reason why the EU exists or also a purposeful misinterpretation. Countries joining the EU are requested not to join because of the single market, but because they want to join the political project of a more peaceful and an improving Europe.
At its core the EU is a project to bring Europe together, to cooperate instead of being in wars (... WW1, WW2, Cold War, ...).
The EU is even more a political project as it is an economic project. But there are many other projects. The market is not the EU, it's a large and important project, the EU can't be thought without it. But there is more. The main motive behind all that is the will to work together. Even the "single market" is at its core a political project.
We have seen it with Brexit. The UK never bought into the "political project". UK politicians also mislead the public about that. The EU is as much about social standards, political standards, environment, competition, building up the countries, joint research, joint infrastructure, a common law, common currency, being competitive in a larger world, ... and so on. During Brexit it was promised by some UK politicians for a short time that the UK would leave the EU, but not the single market. Misleading the public that this was a topic of economics, while the EU really is a political integration project, not just a project to build a single market.
France and Germany at war for many centuries, now have a common integration project (in a larger EU), free movement, no physical border, an EU passport, a single currency, joint projects, ... these are the things which moved us forward.