But that's what I'm saying, it's not the force from the contact patch that leans the bike, it leans despite the force from the contact patch.

The situation at high speed is fundamentally different from the situation at low speed. Try this experiment (this became quite long, but I wanted to make the explanation as explicit as possible to avoid misunderstandings):

1. Put a pebble in the road as a marker and ride towards it quite slowly, like jogging speed. As you approach the pebble, initiate a sudden countersteer to the left. What will happen is that the handlebars will turn slightly to the right, the contact patch will move to the right, and the bike will roll to the left as it rotates around the CM. Your front tire will likely pass to the right of the pebble.

This is the situation you describe: at low speeds, the force from the contact patch makes the bike rotate around the CM, which initially moves opposite to the direction you are trying to go. (It has to, the contact patch force is to the right, and that's the only lateral force affecting the bike initially.)

2. Now try the same maneuver at quite high speed. (Do it in a wide parking lot or something so you don't run off the road, obviously...) What you will find is that unlike in the first situation, when you countersteer, the handlebars are pretty much rock solid. Even though you attempt to turn them to the right, there is no appreciable motion. Because the wheel doesn't turn, the contact patch continues basically a straight line. However, the bike still leans over and starts turning to the left. You'll now find that the wheel will run over or pass slighly to the left of the pebble.

This behavior is impossible to understand without the gyroscopic effects. If the bike still rotated around the CM, the contact patch would still move to the right, actually by a larger amount if you think that it takes a larger force to overcome the stability of the spinning wheels. If that was the case, the front wheel would still pass well to the right of the pebble, but it doesn't.

What is missing in this case is the effect of gyroscopic precession in a fixed gyro. With the front wheel spinning rapidly, the attempt to turn it right when countersteering will not rotate it around in the steering direction, it will cause the wheel to rotate orthogonal to the applied torque, in the roll plane, taking the whole bike with it. If the bike was suspended in the air, it would just rotate around its center of mass, but because friction prevents the contact patch from moving laterally, there is a resulting force with a magnitude that keeps the contact patch stationary. Note that unlike in the slow-speed case, where the force from the contact patch acts to the right, it now acts to the left. The resulting motion is one where the bike rotates around the contact patch, moving the center of mass in the direction of the lean. This motion looks quite "magic" if you're not aware of what the gyroscopic precession does.

Without the precession, you wouldn't need to go very fast until it would be impossible to turn the bike at all. If you just imagine that the gyroscopic effect makes the bike resist leaning over, you'd need to develop a larger and larger contact patch force to the right as you went faster and faster. This would make the bike track further and further to the right as you attempted to turn to the left, which is exactly the opposite of how a bike really behaves.

Hope that explanation helps.