My half ton has completely stock suspension but with 15" radials, and it does not exhibit this issue. I have completely redone the king pins, tie rod ends, and spring shackles and bushings. The spring shackles make a bigger differance than you would think. Don't neglect the rear end, either, the spring eyes and the motor mounts matter if you still have a torque tube. They are what locates the axle in respect to the frame and any wear gets amplified by the springs and turns into bump steer and wandering. Except for the stiffness mine steers quite well. It's a bit numb feeling, but it goes exactly where I point it and it handles bumps pretty well. The best upgrade you can do is a set of sway bars and good shocks.
I'm picky, too, my daily driver is a Mazda RX-8 sports car.