Regardless, I suspect they won't do it because it would be irrational. Of course they have every incentive to make threats. But actually going through with them would hurt them just as much as the UK. Trade is a mutual benefit after all.
Anyway it just strikes me as mean-spirited. I doubt the EU leaders really care about immigration that strongly. They are also holding the UK to a different standard than other non-EU countries, and there is no justification for that.
I'll believe their serious if they start demanding the US, Australia, etc, open their borders or lose trade. Good luck with that. Yet somehow they find it weird the UK doesn't want unrestricted immigration.