But if the airline, or whoever, has their MAIN business office registered in an EU country, then it would apply.
Ah, but now you are bringing the registered office address of the corporate owner into it
See - it's not simple.
It is quite bizzare that if United Airlines set up an EU company for managing the maintenance of the part of their fleet which flies to the EU, they would apparently fall foul of this reg. But the regs make no exceptions for e.g. AOC holders which would be the obvious thing to do. In the pre-EASA scenario, getting an AOC meant that you could do just about anything that was approved in the CAA approved manual - including using an N-reg for charter work, etc. But not anymore.