You are very perceptive to note that for a given angle-of-attack, a return to the exact airspeed that allowed level (horizontal) flight would indeed imply a return to level (horizontal) flight. But that's not what we expect to happen when we reduce the power. Hopefully this answer will help you to understand why not.
When you elevator trim to fly unaccelerated straight and level flight
(all forces balanced) at e.g. 100kts, you are relieving the pressure
needed to keep your aircraft in that angle of attack that produces
enough lift to counter the weight, at that particular airspeed
(coz any increase or decrease in airspeed also will affect lift, so
speed will affect the amount of lift that needs to be created by the
AOA to counter the weight so as to stay level)
Technically this is not quite right. The amount of lift that is required for the flight path to stay level rather than bending up or down is the same at any speed, but the AoA needed to create this lift varies with airspeed. Maybe that's what you actually meant to say, or something close to it.
So if your speed starts to drop, the amount of lift that your AOA must
make = increased.
It may be that the word you are looking for is not "lift", but "lift coefficient". Increasing the AoA increases the lift coefficient. Lift is proportional to lift coefficient times airspeed squared. To maintain horizontal flight as we slow down, we need the same amount of lift, which means we need a higher lift coefficient, which means we need a higher AoA.
Or maybe you simply meant to say something like "if we hold angle-of-attack constant and decrease airspeed, then lift will be decreased, which means lift will have to somehow be increased again to bring things back into balance." This is exactly true, regardless of whether that "somehow" involves an increase in angle-of-attack and lift coefficient, or a return to a higher airspeed at the same angle-of-attack and lift coefficient that we had before.
But since you have trimmed elevator to that position means the AOA is
Yes, at least to a first approximation. We'll assume that is exactly true for the purpose of this answer.
So if your speed starts to drop
The speed is dropping because we've reduced power and thrust, so drag is greater than thrust, temporarily causing the net force acting along the direction of the flight path to be non-zero.
the amount of lift that your AOA must make = increased.
Yes, the drop in speed has caused a loss of lift, and lift must somehow be increased again to bring things back into balance.
In the meantime, lift is temporarily less than weight. There is now a net force acting perpendicular to the flight path (i.e. downward), so the flight path bends (curves) downward.
Since the angle-of-attack stays constant, as the flight path bends downward, the aircraft must pitch down.
As the flight path bends downward, gravity (the weight vector) gains a component that is partly acting parallel to the flight path in the forward direction, and this is what drives the increase in airspeed till it is approximately back at the initial value.
As the airspeed increases, the flight path stops bending down. Actually in real life we are likely to see the airspeed slightly "overshoot" and then decrease again. As the airspeed "overshoots" it provides the excess lift needed to curve the flight path up toward something closer to horizontal flight-- but not exactly horizontal. Since you've reduced power, you'll end up in a descending glide.
then your speed will be back to what it used to be and your that
trimmed AOA will continue to be able to supply that exact amount of
Since you've reduced power, you are now in a descending glide. To a first approximation, your statement above is true. It is a good approximation for shallow to moderate glide angles. But if you want to know the full truth, it is that the lift vector is slightly smaller in a glide than in level flight, which means the airspeed is slightly lower in a glide than in level flight at the same angle-of-attack. For more, see the vector diagrams in this related answer. But you really don't need to understand this to answer your question. The most important thing to understand is that being in a glide rather than in level flight is the only way we can accommodate the fact that drag is now greater than thrust. Again see the vector diagrams in the linked answer-- just mentally substitute "drag minus thrust" for "drag" in the vector diagrams and you'll get the idea.
which tgt with the lift at that same airspeed is just right to
counter the weight. So the aircraft will be back to level flight
No, it will be in a stabilized glide. The weight will be balanced by the vertical component of lift, plus the vertical component of (drag minus thrust). See the vector diagrams in this related answer for more; just mentally substitute "drag minus thrust" for "drag".
Since power is lowered, I would expect the aircraft to continue to
lose speed and the whole cycle of pitching down then back to straight
and level will go on and on.
You are starting to touch on the question of the pitch "phugoid" oscillation but that really has nothing to do with the core of your question. The most important thing to understand is that reducing the power while holding angle-of-attack constant-- or while holding airspeed constant -- technically both cannot happen at once but don't worry too much about that-- will allow the airplane to end up in a stabilized glide, with the weight supported by the vertical component of lift plus the vertical component of (drag minus thrust).
A couple of more notes in closing--
First, this business about the aircraft tending to trim to a slightly lower airspeed in a glide than in level flight, if angle-of-attack stays constant -- you'll rarely be able to detect this effect in actual practice. For example, in an airplane with a propeller on the nose, if you change the power setting without touching the yoke or the elevator trim control, the change in the propwash over the tail will often cause some amount of change in angle-of-attack, which will drive a change in airspeed. In jets, offset thrust lines can cause a similar effect. These sorts of effects are likely to completely dwarf the change in airspeed caused by the fact that for the same angle-of-attack, the lift vector, and therefore the airspeed, must be very slightly smaller/lower in the glide than in level flight. Note that in the vector diagrams in the answer linked above, the lift vector is really not very much smaller than the weight vector as long as the glide angle is not too steep. And since lift is proportional to airspeed squared, only a very small reduction in airspeed would be needed to cause that amount of reduction in the lift vector while holding angle-of-attack constant. Still, when the (drag minus thrust) vector is not zero, it must be accommodated in the vector triangle as shown in the linked answer, which means that lift must in fact be a little less than weight.
And second, if you make your power reduction very gradually, the aircraft will stay close to a steady-state condition. You'll smoothly transition to the final steady-state glide and final airspeed (which for all the reasons noted above, may or may not be a little different from the airspeed you had at the higher power setting). On the other hand, if you abruptly make a large power reduction, you'll see the airspeed drop significantly, and then the nose will pitch down quite far, and then the airspeed will "overshoot" well above the final steady-state value, and then the nose will rise above the final attitude for the glide, and then the airspeed will "undershoot" the final value. This is the pitch "phugoid" in action, and you might see several cycles before everything settles down into the steady glide.