If I'm in contact with the top of my bike and not steering the bars, I can either push the top of the bike left or right. I push the top of the bike to the left. The bike and I have separate moments of intertia around longitudinal axis. Without any other forces involved (just my bum and the top of the bike pushing each other), the bottom of the bike i.e. the contact patch would move to the right. However, there's friction between the tyres and the ground meaning that doesn't happen. Instead, the ground pushes the bottom of the wheels to the left. This means the only external force on the union of rider and bike is from the ground pointing leftwards. This pushes the COG to the left.
While the motion of the top of the bike may cause some countersteering in the front wheel if left free, you can resist this by pushing the right side of the handlebars i.e. steering left. So we now have a bike tilting to the left, with the front wheel pointing forwards, and the COG to the left of the contact patches. The front wheel can be gently pushed to point left.
No countersteering involved.
With counter steering, it can be made much quicker and in a much responsive fashion.
There is nothing magical about countersteering, it’s just the way cornering works with bikes.
Don’t believe me? Set up an angle indicator and video it.