Is the Push Basic Stroke the correct for chipping? Should all three strokes (push, pitch and punch) be practiced with all three hinge actions?
Push Basic Stroke may be used for chipping, and is actually a very good motion to use around the greens! Almost everything except your right elbow piston is zeroed out, and it makes a very consistent motion. In TGM, you always have options, and Push Basic Stroke for chipping (short/medium length chips) is a cool option.
Push Basic Stroke does not produce a lot of power. So confine it only to putting and chipping.
Are you Hitter or Swinger? If you're a Hitter, practice all three Hinge Actions using Push Stroke, and if you're Swinger, use Pitch Stroke. Make sure you're using Single Wrist Action with the Push Stroke, and Standard Wrist Action with the Pitch Stroke.
Also, because it is very unlikely that the Push Stroke is part of your Basic Procedure, I would recommend not devoting 100% of your time practicing chipping using the Push Stroke. Your chip stroke is your Basic Motion, which should be a simple/short version with the same underlying components of your Total Motion. So as your Basic Motion improves, so will your long game!