The problem is the EU has its own serious problems which could have been avoided had Britain been allowed to have a more central role even back when it was the EEC. People forget that we once
wanted a bigger role and de Gaulle did his best to shut us out. The result was growing Euroscepticism which is seemingly being validated by the financial situation now. We were effectively shut out of the cool kids club and reacted against that sentiment (the populace, not the guvmint) by becoming increasingly cynical about it.
Though I'm not sure why you would think the 7th largest economy in the world would only have 'slight' relevence without the EU. The EU isn't the only option, either, the Commonwealth could be transformed into a similar economic bloc with some effort, an idea India is amenable to in order to help balance Chinas influence.
I used to be extremely pro-EU, now I'm not, largely because of supremely idiotic decisions like letting Greece and other second tier economies adopt the Euro. That didn't work out too well, but criticisms at the time were written off as snobbery and Euroscepticism.
Let's not turn this into a EU debate thread, though.