I think that we will have a more representative democracy. Historically speaking if you look at states with large population size like China or Imperial Russia or the USSR they have been unwieldy, difficult to control, have normally involved quite a degree of oppression and have found it incredibly difficult to look after the interests of disparate groups of people fairly.
The USA has done it well, but it's still much smaller than the EU. And they have developed working coherent systems which give a great degree of autonomy at state level whilst still maintaining a cohesive whole. The EU system is nowhere near as admistratively effective, cohesive or representative as the US system and they appear to have little interest in becoming so. It doesn't seem to be an organisation which is interested in improving or making itself more attractive to either member states or their citizens but rather one which seems to be determined to maintain itself with all it's faults by a system of threats, blackmail and financial control. The renegotiations showed it offered little flexibility or recognition towards individual states needs. I'm not comfortable with being part of that.
Also, looking at how the UK struggles to reconcile it's disparate parts and the interests of a pensioner in Glasgow, a lone parent in MW in the north east and an affluent southerner for example - well I really doubt an organisation the size of the EU can really look after the interests of a fisherman in Spain, a working Mum renting in the north east, a farmer in Italy and a Roma in Romania with anything like equality or fairness.
I would like to see a reduction in migration as it's not sustainable at current levels. Nobody has offered any sort of coherent plan for how we will deal with a population growth the size of Iceland per annum without it affecting the very poorest in society in terms of wages, employment, housing and services. I would rather vote for a possibility of change on that level rather than vote for a definite no to any change ever - which is what a remain vote was.
The tube strike analogy is irrelevant, a totally different thing, and I believe something that particular electorate (union members) weren't very happy about.
Perhaps a better analogy might be the last EU elections in 2014? The turnout for that was 35.6% of the entire UK electorate for all candidates. 28% of those votes were for UKIP on an explicit 'leave' platform. Which means only 25% of the electorate voted for their MEPs. By your logic that means that the EU parliament has absolutely no mandate from the UK electorate and it's authority shouldn't be recognised in the UK.
But of course, you won't accept that logic, because it's not actually logic you want is it? You just want to get your own way, nothing to do with fairness or democracy - just a desire to force your will on people who disagree with you.