So I'll repeat the basic premise here:
As said, some Democracies are 'direct' - people can get a petition going, and if they get enough signatures, it can be put to the vote, and a majority will win, even if small. It is actually quite rare- Switzerland, my birth country, is one.
The UK is not, and has not been for many Centuries- That is the absolute backbone of UK Democracy- that a/ Referendum can only be advisory (and NO PM has any right to promise it could be anything but- by OUR OWN LAWS...) and that we elect MPs (with a not at all democratic system of First Past the Post system- which means anyone who lives in an area not representative of their views or politics know that their vote is null and void and will go straight into the bin- so not considered Democratic by most other countries) - and that those MPs, in Our Sovereign Parliamentary Democracy, will represent us and make the Laws and major decisions on our behalf (with the help of a totally Undemocratic second Chamber, the House of Lords).
THAT IS OUR SYSTEM - like it or not- think it fair or not, believe it democratic or not. And insisting on by-passing it for a direct popular decision- is totally against OUR political system. Hence the total contradiction.
If we want to change our system- then we should embark on a massive re-shaping of our whole voting, electing, etc- system. I am, for instance, in favour of getting rid of First Past the Post System, as my vote in the UK, nor my OH's- has ever counted and we knew in advance that it would not- because of where we lived. But then- it is a whole different ball game. Cameron had no right to say 'ah well, let's pretend we are Swiss for a bit, ans see what's what'. Nonsense.
Whatever we think of Brexit. Whether Leaver or Remainer - it makes no difference to our political system:
the UK IS a Parliamentary Democracy, pros, cons, warts and all.