What is the "Centre of Pressure"?
Center of pressure is not a point "where the lift is most concentrated". Center of pressure has very little relation to how is lift distributed along the wing (you can have exactly same centre of pressure for infinite number of different distributions of lift).

The center of pressure is mathematical sum of all local pressure vectors along whole wing surface. But it is only, more-or-less, thought summing, this point has no special meaning in terms of stress distribution or so (at least no straightforward one).

You could as well continue to put all local pressure points into equation of movement and you would get exactly the same result. But equations with plenty small force vectors would be unnecessary complicated and if you do no need this level of complexity, it is easier to sum all forces in advance and use the summed value instead.

Note that if you would want, say, for example, calculate stresses inside the wing, you can not do like this. You would be forced to put each local pressure point into the analysis. But as long as you could assume whole wing to be solid (which is usually acceptable approximation for lot of simple problems), using single summed lift vector is easier.

If you simply sum all fractional pressure vectors, you get total vector of aerodynamic force. Which is enough to know how big force and in which direction acts on the airplane. But in order to be able to analyze rotational effects, you need to know in which point this total force acts (because same force vector shifted into different point of solid body will cause different rotation). And this is where center of pressure comes into the play, as it is the point where thought summed aerodynamic force should act in order to affect airplane (as a whole) by same translational and rotational effects as actual pressure distribution along the wing does.