This could be easily solved, if the burden of proof was on the company itself... when they report/sue you for copyright infringement, they'd just have to prove, that it's reasonably simple for the person to obtain the material legally (so, is there a (eg nintendo) distributer in the country of the accused, is there a store that sells those games there, is it possible to buy it online, etc.). If there are no reasonable legal ways to obtain the material, then you cannot claim lost profit and the "pirate" cannot get fined/sued for infringement.

Seems unlikely that it’s a particularly hard problem to solve, given that trademark law works in a similar way, where if you stop actively using the mark in the market, you can lose it.

Why? Make copyright something you register and pay for and companies making no money from them would simply stop paying.