" The MPs are (supposedly) elected by democratic process and - ideally - should vote on things according to the wishes of the majority of those who elected them [...]"
Westminster is a "representative democracy". An MP is elected to make the best call on an issue according to their considered judgement - even if the majority of their constituents have a different opinion.
If the "Leave" vote had reached the usual constitutional change threshold of 66% then it would have been more understandable.
Always acceding mindlessly to the opinion of the majority in the population is a recipe for mob rule and the oppression of minorities.
That the MPs mostly just rolled over after the advisory referendum was a dereliction of their parliamentary duty.
The USA Founding Fathers created the presidential Electoral College with the mandate specifically to veto the election of an obviously inadequate populist candidate. That safe-guard also failed this time.