The FBI warnings are about public playback and illegal copying, both of which are explicitly outlawed by copyright law itself.

Mere execution and usage of software is simply not something copyright law cares about, which is why your assertion that

> Running software without a license is literally illegal (at least in the US).

is wrong. There is nothing in the law that gives copyright owners an exclusive right to "usage", only to the making of new copies, their distribution, derivative works, public performance, etc.