I believe that for the elevator to see no net force it needs to be aligned with airflow across the stabilizer, so it makes sense that the neutral coordinate system for the elevator is with the stabilizer. The problem is that changing the trim changes the trim speed of the plane, so if there was elevator used to compensate for a mismatch between the current trim angle and the correct angle for that speed, then for level flight with changing airspeed the pilot would either have to add or release a load on the elevator which means a change in stick position. OTOH, if the stabilizer trim is coordinated with the speed change, then there would be no need to even touch the control wheel, which would remain at the neutral position. In all other cases, where the trim was incorrect for the airspeed, then correctly trimming will allow the pilot to release the load on the control wheel whereupon it will return to the neutral position.
If the wheel is left free, then changing the air speed or moving the trim from the trim-speed position will cause the elevator to move relative to the stabilizer and the control wheel will move with it. I don't recommend testing this in flight; instead hold the wheel position and feel the change in force as an indicator.