Saying that the public are in control of the direction of travel but leaving the experts in charge of the implementation (as long as it doesn't go against the wishes of the public, i.e. politicians can't just choose not to invoke Article 50, but saying they're free to explore constructive EFTA options) seems preferable.
As I see it, it doesn't really matter if we agree that EFTA is the moderate brexit option: the primary focus of the Leave campaign was on freedom of movement, and it was clear (even before the referendum, given the situation with Norway and Switzerland) that freedom of movement is an integral part of the single market treaties -- a point reiterated by Merkel and Tusk over the past few days.
So regardless that you and I may prefer for the Leave side to explore the EFTA route, it isn't in their mandate -- they won on a campaign of curbing free movement. Had the "moderate Brexit" been a separate choice on the ballot, they might have had a mandate to go for that option. But right now, nobody really knows how many of the Leave voters voted against free movement, and how many voted "just" against the EU.
The politicians making the promises may not even be in the government (it tended to be Leave.EU and Farage that focused on immigration, Vote Leave was wary of doing the same and focused on the economy). All politicians can do in a referendum campaign is talk about implied benefits one way or the other.
It's now up to the politicians to work out the best deal for our country. It's going to be a much smoother ride (and much better for the country as a whole) if free movement of people is allowed and we join EFTA than if the government tries to hold firm on that one, for sure.
Thinking about that last option some more, maybe it would make sense to put a nuclear option (like "destroy chunnel") on the ballot, to weed out the spiteful votes from the constructive ones. Even if you explain beforehand that the chunnel will never be closed -- some people just need to vent.