The biggest deal to me is people knowing that they push left to go left, push right to go right.
After that it is your choice how conscious you keep it in your everyday riding.
Against the "just ride" argument are the cases where people have been getting by for many years with steering by "leaning I that direction" (causing them to push-steer without knowing it). That works right up until the time that you need to maneuver really drastically. No amount of leaning is going to save you then.
THAT is when you need to know that you push hard to go in that direction.