I do not deliberately fly aircraft out of trim, though if the maneuver is dynamic, I won't retrim all the time either. The trim system is not provided to fly the plane, it is provided to relieve the pilot of prolonged control forces during stable flight.
Control forces are a vital element of our being able to fly the plane. Without the feedback of a change in control force, and the increased control force as G increases, pilots would be having a terrible time flying with a failed ASI, and tearing the wings off with high G in tight turns. The design requirements for certifying the aircraft specify that in some conditions control forces increase, and not reverse.
I've flown a couple of deficient aircraft, whose elevator and/or rudder did not comply with these requirements, and they were very challenging to fly safely in certain conditions.
Tab or all moving H stab will produce more positive trimming results than a spring system (which is my distant recollection of the PA-38). But none are expected to be so precise that the pilot no longer has to do some flying!
A properly loaded certified plane will not run out of pitch trim in normal flying configurations - if it does, something is wrong. My experience is that Cessna 182 and 206 can run out of nose up trim just a little, while being glided at forward C of G.