Pushing forward with the firing hand makes sense when you don't have a stock to work with and the recoil you're dealing with is relatively mild.
Pulling a stock into your upper body makes sense when you have an actual stock to work with and real recoil to deal with.
Pushing forward with your support hand makes sense when you are running a pump gun because hanging force rearward on the forend of a shotgun just prevents the action from unlocking.