In my experience many rockets are mildly unstable and need to be actively held on the prograde vector during the gravity turn. That said if you have a sufficiently stable rocket, which is possible with fixed fins, and have enough control to do the initial pitchover (or put the rocket on the pad at an angle) then you can fly the whole ascent hands-off. Even then, though, a good way to make the initial pitchover is with a gimballing engine.