You have to separate two concepts; static stability and trim forces.
The weathervane analogy works well for this when conceptualizing static stability. A weathervane points into wind when its aerodynamic center, the point at which all forces induced by the wind is focused, is downwind of its pivot axis. Wind shifts, it points into the new wind direction. It's statically stable; it wants to always point into the airflow.
If I move the pivot axis of the weathervane back to the same spot as the aerodynamic center, it won't point any longer and will just drift. Its static stability is neutral. If I move the pivot axis even farter back, it becomes statically unstable in the original direction, and will seek stability in the opposite direction. It'll switch ends.
Then there's trimming forces. If the weathervane is flat, it points into wind along its axis. But say I want it to be statically stable, but point at an angle off the wind, so that some side force is generated on the pole holding the vane up. I bend the trailing edge to one side. I'm trimming it. The local force created by the bent trailing edge will drive it out of alignment with the airflow until the opposing forces of the wind acting on the rest of the vane reaches equilibrium.
It will now be statically stable, and will point consistently at some angle off the wind depending on the amount of trimming force I applied. I could call that Angle of Attack. So trimming forces modify the angle to the flow at which weathervaning forces (static stability forces) want it to point.
Turn the weathervane axis 90 degrees to horizontal. Same thing is happening. It wants to point into wind in the vertical plane, and trimming forces can influence the relative angle of the body to the wind at which it points, in the vertical plane.
Now take away the fixed pivot, so the weathervane is a free body. The pivot axis now becomes the weathervane's center of gravity. If the aerodynamic center is still behind the C of G, it will still want to point into wind. Absent any trimming forces, being free in space, if I throw the body, it will go in a very statically stable ballistic arc into the ground as it's drawn by gravity, maintaining alignment into the airflow the whole way. Like a lawn dart.
So to make it resist gravity and fly, I have to put lifting foils or wings on it, while keeping the aerodynamic focus point where I want it behind the C of G to keep it statically stable, and trim it to point into the air at an angle that makes the lifting foils operate at a positive angle. Now when I throw it, the body wants to align into the wind with the wings at a positive angle, and if the angle is right, enough lifting force is made to resist gravity and it flies straight instead of going into a ballistic arc.
On the wing, Center of Pressure becomes just one of the various influences that determine where the overall aerodynamic center's point of focus results. Because we need the wing to lift up, we need to apply trimming forces in the direction that forces the airplane to be statically stable - to vertically weathervane - at an angle to the flow that generates an up force. The trimming force therefore has to be opposite to the desired lift force, so it's down. But for stability purposes, the tail's force orientation is as part of the overall weathervaning effect, whatever it takes to maintain the trimmed angle of attack.
The tail is actually doing two separate jobs - contributing to static stability by adding surface area aft of the C of G, and generating trimming (and maneuvering) forces. Like a weathervane turned horizontal with the trailing edge bent up, it generates a down force to trim the aircraft to seek an angle to the flow that angles the wings into the flow to create lift. It also adds its area to the overall footprint of the whole aircraft that determines where the aerodynamic forces in the vertical plane will be focused, which must always be aft of the C of G (static margin). So that when the angle to the flow changes from an external source, being statically stable in its trimmed angle of attack, it will seek to maintain that angle.
Pitching moments, Centers of Lift, etc, are just sub-parts of the set of forces that result in the aerodynamic center, or Neutral Point, at any given time. If the aerodynamic center or NP, the sum of air forces acting on the fuselage, wings, nacelles, and tail is at x, while the C of P is at y, and the C of P of the wing moves aft, the overall aerodynamic center moves aft a little bit, increasing static margin, while also increasing the trimming force required to hold the same angle of attack. And vice versa when the CP moves forward.