> Surely the best measure of whether a politician speaks for the people is whether he or she is elected.

There’s more to it than that. For example someone could be elected though not desired by the majority (as has happened in US presidential elections a few times, and also happens in various places where more than two candidates vie for a seat).

Also there Re situations where the majority will egregiously violates the rights of minorities.

Or when there is a muddy signal, where Brexit is a good example: some people wanted any possible Brexit, others wanted it assuming certain conditions that are now clearly not going to happen; what is the popular will in this case?

I’m not saying that being elected isn’t the most important element, but it’s not only not sufficient, but it can actually be wrong.