We could go round in circles arguing about the actual role of an MP. As far as I know, it isn't written in stone, so everybody interprets it differently. MPs would be foolish to vote against any known consensus in their constituency because they'll lose their seat, but there isn't any obligation to do what their constituents want.
MPs represent their constituents, but they're not there to do the populists' bidding. They are supposed to have an overview of the bigger picture and to understand the national and international context. A bit like responsible parents, I suppose.
Some people will always vote for the same party (less so now, I think) for whatever reason, while others might vote for a good constituency MP, who spends time in the constituency and takes the time and trouble to act on individuals' concerns.
To be honest, with a FPTP system, it doesn't matter how most people vote because only a few hundred thousand people in marginal constituencies make any difference. Personally, I think that's a big part of the problem. People feel their voices aren't heard and one thing that the referendum did was give people a direct voice.
There were many constitutional "experts" at the time who questioned how it would work. They pointed out that it was never going to be as simple as just walking out and starting out on our own. Unfortunately, the sensible voices were drowned out by the hysteria and false promises and people assuming they knew what would happen because that's what they wanted.