Lots of people wanted it to become a state. The ideal of a European state as a way of preventing war in Europe dates back centuries, and has been regularly proposed again. The most serious recent proposal being Winston Churchill who specifically argued for a United States of Europe after the war.

Churchill's efforts were a major factor in the establishment of the Council of Europe (not EU), and the European Court of Human Rights (CoE; not EU) and what became the EEC/EU.

So while many people involved also did not want to go all the way, historically the origin of the EU includes a lot of people who went into it with the explicit goal of eventually turning it into a state.

A core principle ever since the Treaty of Rome in 1957 has been to create "an ever closer union". You can't do that without eventually ending up with a state.

Heck, the Holy Roman Empire was basically that. From a modern point of view you could say "but it's basically just Germany" but Germany wasn't a thing back then. Except of course the Holy Roman Empire was a monarchy.

As for Britain's involvement in establishing the Council of Europe, I've heard Brits joke that Britain created the EU to keep the continent busy, not to be a part of it.