Is a trimmed flight not always an unstable system?
But assume, there comes a wind gust, and the angle of attack will
increase. This causes the center of pressure to move forward in front
of my gravity location. So the aircraft will become instable because
the angle of attack will further increase (nose up). The only thing
the pilot could do is to use the elevator to trim in pitch.

This is not accurate.  Are you familiar with &quot;free-flight&quot; model airplanes?  Some of them are stable enough to be flown in the middle of the afternoon when the atmosphere is full of thermal updrafts and downdrafts, with no input from a pilot whatsoever, and no automatic control system (autopilot etc).
You are correct that if some disturbance increases the angle-of-attack of the wing, with a conventional airfoil with no &quot;reflex&quot;, the center of lift and center of pressure of the wing will move forward, which is destabilizing.
But the tail is also seeing an increase in angle-of-attack, or if the tail is set to fly at a negative angle-of-attack in normal flight, then the angle-of-attack of the tail becomes either less negative or somewhat positive due to the disturbance.  Regardless of which of these is the case in any given instance, the change in angle-of-attack of the tail, which is far from the CG and therefore exerts lots of leverage, causes the center of lift and center of pressure of the whole aircraft to move aft of the CG.  So the aircraft noses down to a lower angle-of-attack, where the center of lift and center of pressure are again co-located with the CG.  That's how pitch stability works.
For more, see this section of John Denker's &quot;See How It Flies&quot; website.
Note-- for simplicity, I'm assuming that the only forces we need to consider are the &quot;upward&quot; or &quot;downward&quot; ones.  I.e., that the &quot;center of lift&quot; and the &quot;center of pressure&quot; are one and the same.  This is not exactly true.  For example, if the thrust line were way above the CG, then it would still be true that the center of pressure (including the effect of the thrust vector) was co-located with the CG when the aircraft was at equilibrium, but the center of lift would have to be a bit ahead of the CG.  If you find this simplification to be objectionable, then just (mentally) strike out &quot;center of lift&quot; from my answer and just leave &quot;center of pressure&quot;.
In my answer here, I'm assuming that some sort of disturbance, such as a sudden updraft, has increased the angle-of-attack of the aircraft.  However, there's actually an additional question embedded in your question, which is &quot;will a purely horizontal gust of wind cause a change in the angle-of-attack of an aircraft that is flying horizontally?&quot;  That's a whole other &quot;can of worms&quot; that we may not really need to get into right now, but let's at least &quot;lift the lid&quot; and take a peek.  The answer to that appears to be that the sequence of events is as follows:  gust of wind from front -&gt; increased lift -&gt; upward curvature of flight path (which may temporarily decrease the angle-of-attack slightly) -&gt; nose-up change in pitch attitude to conform to new direction of flight path -&gt; continued upward curvature of flight path and continued nose-up pitch rotation.  All assuming that the aircraft has not yet come to equilibrium with the new velocity of the surrounding airmass at any given point in the chain of events.  It's not obvious that the angle-of-attack is ever increased above the original angle-of-attack.