There's three possible paths: tweak the way the EU works every time a problem arises, scrap the current EU and go back to mini-states, or scrap the current EU and go full integration. Right now the EU is still trying to operate as a confederation, and those are rarely stable in the long term (Switzerland being again the exception).
Do you think it would fly in the US if a member state decided that the constitution doesn't apply to them? Or threatened to leave the Union if you didn't grant them an exception on the first amendment, or ultimately if you didn't let them rule the whole union? Because that's ultimately the issue the English have with the EU: they aren't the ones ruling it unlike the UK.
Personally, I'm glad that the UK finally took a decision. For decades they've literally been EU's cat, always on the wrong side of the door. They've been threatening to leave for ages, in order to get exceptions allowing them to take the full benefits without taking the full responsibilities or following the same rules as the rest of Europe. They can go wallow in their pipe dream of a British Empire restored, while the EU can continue on its own path without the added weight of the crazy ex-girlfriend that breaks up with you, publicly insults you but still wants to benefit from being in your house and raiding your fridge without paying her share of the rent.
The EU has been clear from the day after the BREXIT vote: even tho it pains us to lose a long time friend and ally, the UK has to respect the will of its population and the sooner the better. The EU isn't the one stalling here, the UK is. The issue is that the UK government knows it's going to be a shitstorm if they trigger Article 50. They're fucked if they do and they're fucked if they don't, the only option that makes sense for them is to stall as long as possible so it becomes "someone else's problem". Bonus points if they can stall it until after the next election, so they can have a clear idea of what the population really wants
I hope that BREXIT will end up being a net positive for the EU by forcing them to have a real serious discussion on where they want to go post-BREXIT. It's a shame it will have to make life so painful for a lot of my friends.