The purpose of a representative democracy ought to be pretty obvious, because not every decision can be voted on by everybody. That would be a waste of time and money, and would defy common sense. No, representative democracies make much more sense than you give them credit for.
Representatives are accountable to the people for the decisions they make, but are then left free to get on with it while the people are free to live their lives in the meantime.
With respect to the EU vote, then, MPs have been given an instruction, and it is up to them how best to carry it out. That may involve accepting this deal, or it may not; and if they cannot reach consensus, then they may wish to ask the people again. In all cases, democracy survives. It's a complete fallacy to pretend that it dies as soon as MPs start thinking for themselves.