There's also some amount of national politicians laundering unpopular at home legislation through the EU, having it proposed and passed there and then throwing

There's also some amount of national politicians laundering unpopular at home legislation through the EU, having it proposed and passed there and then throwing their hands up and saying "See, it's the EU making us do this". The UK's "success" on that front is part of why the anti-EU sentiment there was so high.