If it's old dry wood you may want a full depth pilot hole just to keep the wood from splitting. There are different sized phillips head and you may not have the right size. but it could just be the awkward angle, as you mention. If you can set the clutch on your driver, set it lower to get just enough torque and allow it to slip before it starts spinning in the head of the screw. But if you don't mind the look, a square drive screw, as you noted, is probably your best solution.