That's especially true when equally non-binding but current polling of your constituents on the same question shows that the opposite side quite likely leads now that the concrete options are known.
It's a well-established British Constitutional principal that a Parliament cannot bind a future Parliament. Why would one view the public will ant differently, even if it were fully informed?
Representative democracy means that they determine policy by choosing representatives. It's disingenuous for those representatives to turn around and blame their own policy choices on an explicitly non-binding referendum that they sent to the public. (Especially so for a leader of the party who choose to do that specifically as a means of minimizing the impact of the issue so referred on a general election so as to preserve their partisan majority.
Then it should have been a binding referendum.
Quite honestly, there is nothing stopping you from doing this anyways. This is the point of democracy. Nothing is ever final.
The one they had in UK was a non-binding one, which can usually only be brought to a vote by the parliament.
So the answer to why anything happened in the UK, including holding a referendum, is basically just: because Parliament decided to.
Parliament votes to have them
> We don't have such a thing in the US.
Well, Congress doesn't choose to have them (separation of powers helps here; the Brexit referendum was very much about electoral strategy in a system where voting for a district legislator is also voting for a particular executive administration.)
> Certain states do, but not the country.
What many US states regularly have is binding referenda with Constitutional force and process; while these may sometimes be of legislative origin, they could do Brexit-style referenda (because of their general powers) but, again, separation of powers kind of eliminates the political use of them, or at least that of Brexit specifically, and a non-binding referenda on a non-profit proposal isn't something that makes a lot of sense outside of very particular political circumstances.