Sorry, the British are not European... they are...insular
Insular? Yes, but European also. In pre-easyJet days there was once a newspaper headline - "Heavy fog in the Channel; Europe cut-off" which illustrates our Anglocentric viewpoint.
But everything we are, our art and architecture, our sacred and profane beliefs, our science and culture, industry and scholarship are all rooted in a common European identity. A learned person can stand in Krakow, Norwich, Florence, Heidelberg or Antwerp and see the same early and undeveloped application of Vitruvius to buildings that heralded the same Renaissance, and likewise the first and second Enlightenments reached across the Channel westwards just as they reached eastwards into Poland. Similarly, you could step into any English church and hear the people proclaim the same Nicene Credo as heard in every Catholic church from Porto to Poznan. Our arts and culture, music and painting fed from each other in an interchange of ideas and innovations. From the earliest days our trade with the Hanseatic ports and the Baltic brought our wool to clothe the szlachta and returned with iron, resin and furs. So we're absolutely European in this sense.
But it's true we're most reluctant to give up our sovereignty to the EU. Perhaps it's because no-one has invaded us in a thousand years, that we have never in living memory been occupied. So we want all the advantages of a free market, of free movement for goods and workers, and we're even happy to pay into EU funds to build infrastructure like roads and bridges and ports that assist trade and movement in the newly-joined nations. But we look at Van Rumpoy and Barroso and shake our heads - the idea of such people governing us is an anathema.