If shifting your center of gravity was all that was needed, how are you supposed to turn a motorcycle a low speed without tipping the bike over when you're leaning into the turn?
This video demonstrates rather clearly: https://www.youtube.com/watch?v=QNRJNwXljUo&t=285s Guy's head is upright the whole time. This rider demonstrates it as well https://www.youtube.com/watch?v=QUUY5xn2p2s
You have to keep in mind how counter steering works. Take a pencil and try to balance it upright in your hand for a split second, then jerk your hand to the right. The tip of the pencil falls to the left, correct? Like wise, if you're trying to keep it upright, you move your hand into the direction the tip of the pencil is falling towards to get the bottom underneath the tip.
That's what basically happens when you counter steer; the front tire turning is kicking the bike out from underneath yourself as it steers to the right, and it forces the bike to lean left. It's the leaning of the bike's tires that causes the actual turn to the left.
The beauty of it the way the bike's frame and front wheel design, the front wheel naturally will want to turn back into the direction of where it's leaning in an attempt to stabilize itself. You preventing controlling that behavior, along with brake and throttle, is how a turn is maintained and how you exit a turn.
That's not to say that COG doesn't affect the turn rate / radius, it absolutely does. It's also possible to turn the bike by just shifting the COG. But counter steering at most road speeds has a much higher maximum turn rate with better control over getting the bike pointed where you want to go and typically with less physical effort to initiate, maintain, and terminate the turn. And why I urge you to try again with counter steering (not necessarily push turning, you can pull to achieve the same effect). It's something easy to practice every time you make a turn and can very easily be the difference between swerve and a crash.