The reason we don't see a wholly European solution to many of these problems is due to the fact that the decision making progress is functionally biased towards each of the member's states unique interests.
The most important institution we have in Europe is, in my opinion, the European Parliament. It is designed in a way that ensures fair representation of all countries and at the same time enables us to pass legislation for which there is collective support. Its members are elected directly by the people in the European elections, which are separate from national elections, for obvious reasons.
All of this, however, is subverted by the existence of the European Council as well as the Council of the European Union (yes, the naming is a bit confusing). Their members are respectively the 28 heads of state of each EU country and members of their government.
This means that, while the European Parliament is elected with the express purpose of dealing with EU law, the European Council and the Council of the European Union are fruit of very different elections and because of that, they are bound to put their national interests first, since they might lose votes at the national level otherwise.
In my opinion, the European Council and the Council of the European Union should be replaced by a body elected directly by either the European Parliament, or in some way, the European population.