The argument seems very all or nothing, being increasingly visible about interviewing elsewhere might get the employer to take actions that help the OP prepare for a new job.
Citizens of other countries act as liaisons abroad to bring in their network and raise the GDP of their origin country by getting it involved in business it never would have heard of.
When I am involved, I shutdown anything that involves the US and can't be solved with a similar partner in Canada or Europe, without all the hassle.