Is "who decides where and when the aircraft flies" a significant defining feature of an operator? Sure, with an airline, the airline is the operator, it decides where and when the aircraft flies and the pilot flies the aircraft from A to B. Much the same with a corporate flight department. But with a flying school the operator is the flying school, not the instructor. But the flying school doesn't decide specifically where and when the instructor and pupil will be flying. Within broad guidelines that's left to the discretion of the instructor.
When I used to be in a group, I always considered the group the operator of the aircraft. In a similar manner to a flying school's booking procedure, the group decided who had the plane when and the specifics of the flight were left to the pilot. The operator didn't suddenly change each time the brakes came off. If there is anyone who could validly claim to have separate pilots and operator, it must be group-owners.
Having said that, this operator / pilot thing is a very weak loophole, if it exists at all. It seems a lot of effort just to avoid having to put in a little effort studying something which is mostly fun and interesting. It would take a brave person to risk using it rather than simply getting the proper paperwork.