yes i would look into the tire issue then
one other thing to consider is you are still dealing with an electronic controlled steering, it is not truly manual like the old units that used a cable to control the drive unit position so taking that into consideration you may not get an exact split in revolutions in the steering tiller between right versus left from center steering directions. Being it is still relying on the electronic signal from the wheel tiller encoder it may still be getting a signal to move the wheel, but taking the resistance of the drive tire and steer gearbox 'true center' position it is possible that it may not move the wheel exactly as you are turning the wheel/tiller.
My guess is the rubber tire is probably causing alot of your issue mainly because you said it turns fine when the load is taken off of the tire.