" A small price to pay for getting my right to vote on who governs me back, but YMMV."
You seem to forget that you do get a vote on that part of government regulation which is affected by the EU - through your national parliament (which sends people to the European Commission) and through your vote for the European Parliament (which being PR might give you more influence than in a FPTP Westminster election). Just because some folk keep repeating "undemocratic EU & unelected bureaucrats" doesn't make it true.
However, you have little influence over the House of Lords. And the way we are governed is also greatly affected by issues no government, national or otherwise, has much effect on such as the weather, scarce resources and some international conflicts.