I'm sure the pressure is from both. I think there'll be a time where Toyota Japan starts putting measures in place to prevent all the gray market import parts from crossing markets as well.
Our litigious society has made manufacturers shy. When TMS USA sells a part it is inferred that part "meets" DOT, EPA and NHTSA requirements. Many non-US parts would not have the pedigree necessary to do that. They do not want the potential liability. If the item is sold to a known legally-imported vehicle that lessens the exposure. This also infers that the part is not going to be rexported to another market.
And if you are crossing the US with a non-US vehicle in temporary import, would you be able to access the parts?
I guess the reason those parts exist in Toyota US warehouses is to be available for worldwide vehicle that may need it...