Can’t comment on the bulk of your post, as I know quite little about the dynamics of UK politics. But this section is pretty relevant to political activity in all Western democracies. In my opinion, though, I don’t think you have the right of it. “Democracy,” unless you are taking it at a very reductionist and absolutely literal level, is not in modern practice simply taking a vote and that’s that. Even the UK, with it’s hodgepodge of a constitutional setup has a definite structure in place that is democratic, but not pure democracy. The USA is the same, being a federal republic. In both of these cases, and in the case of pretty much every other European state, “democracy” means adherence to a set of broadly democratic principles set within a structure that attempts to minimize the drawbacks of raw democracy, which can become mob rule, without unduly restricting the popular will.
These structures operate democratically, in that they reflect the will of the people broadly construed and sustained over time. They set up processes for taking popular input, whether through representative government, petitions, referendums, and what not, and these processes are in turn subject to broader controls, reflecting the overarching ideology of the nation. These could include constitutions, basic laws, mixes of these and tradition, whatever, but in each case they are part of a system that we call democratic, but which is at no point really a pure majority rules democracy. And this is quite intentional.
A referendum in such a system I would argue does not supersede the rest of the political apparatus. Nowhere in any of these systems, as far as I know (though I’m no expert on European politics, for sure) is there the understanding that in the event of a popular vote on something, all the rest of the system gets bypassed. A vote to leave the EU in this case certainly seems to me to be a strong expression of the will of a large part of the electorate at the time, no doubt, but to say that this in turn should mean the national government ignore all the rest of the systems that govern the UK and turn into what amounts to an errand boy seems to violate the spirit of the entire system, if not its own internal rules.
It seems to me that the process that has ensued since the vote has in fact been democracy playing out as it does normally in the West, through the structures of the government. The referendum required the government to address the issue raised, which it has been doing. It does not require the government to abdicate its responsibilities and simply execute decisions made by majority vote without any other considerations. That would be a recipe for madness. We’ve gone through these things in the USA over our history, and there’s a reason why referendums and simple majority polling aren’t used for most serious decision making here.