It is true that with the name EU, the EU exists only from after the Cold War. However, its precursors, and the goals it embodies, existed for much longer, and the reforms that led to the re-branding of the European projects called EU were

but new members of EU were against it.

Against deeper integration were chiefly Britain ("Old Europe") and Poland (one of the "New Europeans"). The rest of the new members had, on one hand, a more vague position, on the other hand, don't have that much weight.

I know that some of "Old European" leaders like Prodi and Chirac wanted to create inside EU the core group of countries with separate European identity and interests

Separate EUropean identity and interests already exist, the 'core group' would be about deeper integration (a quantitative, not qualitative difference). Prodi merely didn't loudly oppose such an idea, it was more Chirac who threatened with it, but Schröder and Fischer didn't go along.

_{*Lunatic*, n.
One whose delusions are out of fashion.}