Again, I agree with your last statement.
I am bothered by your repeated use of the words 'cherry pick' though, because I am not suggesting the EU will allow the UK "a la carte" access, which is what cherry picking implies. Rather, the main hope of the UK finding a resolution is convincing the EU that its own voters also want the same changes that the UK wants (which I think many of them do) and that the rules should be changed FOR ALL. That is not cherry picking.
If the EU wanted to do that anyway it really should have done it when Cameron was trying to negotiate a new settlement before the vote. The EU cannot ignore the role it played in all this. Of course the UK is mainly culpable, but the EU was its usual stubborn and arrogant self. Speaking as a liberal who values open borders in Europe, the EU should have recognised the concerns that many, many Europeans (and Americans, and others) have about immigration, and addressed them in a meaningful way. But no, as we saw with Cameron, the fact that someone asks for it is reason enough to say no. That is the small thing I disagree on. "EU beurocrats who might have been keen to tweak the rules a bit to make sure it remains in the EU will now argue that they won't accept any reforms." The have already shown they were not willing to tweak the rules. They will dig their heels in now, they dug their heels in when we asked nicely. What do you actually have to do to get what you want, not even ask? No, the truth is the EU is completely dysfunctional. I console myself with the hope that it is only through crises that meaningful change ever happens, and that maybe, out of all this, a better EU (inc UK) will emerge, that addresses the concerns of voters, and becomes more accountable. Either through collapse of the existing EU (which I think is now necessary if something better is to take its place) or through urgent reform that is taken once another member says it wants out, and it realises the writing is on the wall.
Having said that, as I said, I agree that the UK was more influential inside than it will be out. Brexiters feel we are utterly without influence in Europe, because we cannot control it. They ignore the fact that the UK was its free trade champion, and that the single market would not be what it is today without UK influence. For that matter, the EU would probably not have been enlarged eastwards without UK influence. That second statement is far more speculative and debatable, but it is a matter of record that the UK argued for expansion. Making it all the more ridiculous that it was enlargement, and inclusion of weaker economies into the bloc, that created "problem" immigration, which ultimately led us to vote to leave. As the saying goes, be careful what you wish for.