There is no address in a UK passport
In other cases the operator would be the club, company, or other organisation that charters it out to others.
which could easily be a company outside the EU, and let's face it this is a very common situation - even covering flying schools doing self fly hire.
What is so difficult to define that?
Because law is not the same as common sense, which works only for the most obvious scenarios.
the operator is the person or organisation responsible for the aircraft.
Responsible for what aspect of it?
It's not so easy.
I suspect EASA used some pub lawyer in Brussels. He was quite clever, avoiding the standard pitfalls like long term parking limits. But he was probably not involved in aviation to any depth.