As you say, having done everything in their power to undermine the institutional effectiveness of the EU, the Eurosceptics cannot reasonably turn around and blame the results of their own actions on the EU. Nonetheless, that's exactly what they do. It's very similar to the Republicans in the US making sure the federal government can't pass any bills, then blaming 'big government' for being ineffective.
Case in point: The migrant crisis, caused by large numbers of refugees coming to Greece and Italy's wide open coasts. How did they get there? No unified European border or coast guard and no integrated defense, which Britain ensured would never happen.
No government should tolerate this kind of behavior, as it is a hostile action against the state, holding the population hostage. It should be treated as a form of criminal extortion or terrorism.