They can't decide for the US. But they can definitely decide for themselves, for example to allow everyone from the US get into Mexico without checking anything.
Another example. Pakistan may decide to follow every EU law and directive and incorporate it into their laws. They may also decide to have the Euro as their national currency. That's fine. They can do that. Of course, the EU does not have to reciprocate.
What I will say will be very oversimplified as I think you lack the most basic knowledge of how relationships between two countries (or blocks of countries) work
DO you know that UK chossed Brexit because they want to restrict Freedom of Movement?
If a German goes to London and UK applies restriction of movement (Brexit), he will be turned back.
UK can give as much freedom of movement in people, goods, services, etc...is as you say their choice. And they said ALL THE TIME that they will restrict freedom of movement.
As Mexico can't decide for US, UK can't decide for EU. So if UK goes out (Brexit) of the Customs Union, UK is out. EU could decide to give freedom of movement, freedom of goods. feck! is what actually is happening now and the EU is willing to continue like this. But if the UK doesn't want to give 1 of the 4 freedoms, the EU will not give neither the same freedom (is basically, reciprocity), but it happens that the 4 freedoms in the EU comes in a pack, you can't divide them or 0 or 4, you can't cherry pick.
So after Brexit, they can decide which freedom (and how much freedom ) can give to anyone/anything, but the EU will do the same. And without a treaty that states how much freedom of movement you can't have, there is NO MOVEMENT AT ALL. UK is blocked from the EU, nothing can't enter to the EU from the UK if the EU doesn't allow it and viceversa.
The "emegency treaty" after brexit is the WTO (You look it yourself) where the EU and the UK are part of it and is not 100% but a certain amount of freedom at a price (taxes, burocracy, etc...) so people, goods, services, capitals, will have to be controlled and to be controlled you need a BORDER.
But WTO only contemplates a few things. for example without an agreement (probably it will be an emergency one, even verbal one) 1 second after the 29th of March, the UK planes, can't fly over europe and viceversa. And that is just an example. Basically because the agreement that allows planes to fly to europe and viceversa will be burnt after Brexit