I agree this will happen, but I object to part of your explanation...you're attributing the AoA recovery to the increased airspeed. I'm saying it's the reverse. The reason is that the pitching moments aound your CG are only in equilibrium at the trimmed AoA/Airspeed.
If you are even *briefly* successful in increasing your airspeed with thrust, the aircraft will pitch up to restore the trimmed AoA, so the problem will follow you around like gum on the bottom of your shoe.
The reason that your airspeed eventually recovers is that in an effort to maintain the same attitude, you are forced to push or pull on the yoke, which produces an AoA/airspeed change. When these things happen very closely in time, the causal relationship is a bit unclear and perhaps not meaningful, as long as both actions *do* happen.
BTW, this isn't just speculation on my part....I've got the formulas in dozens of books which show how to calculate the equilibrium AoA on an aircraft. Thrust only enters into the equation via its distance from the CG, due to the pitching effect that I mentioned. If the distance is zero, then thrust has no effect.
I wouldn't call that a deep stall, which is something else, but agreed, you cannot power out of this, because the stall is caused by excessive elevator. As long as the elevator remains in the same position, the stall AoA will persist. Again, assuming no pitching moment produced by thrust.
In something like the CRJ, I understand there is a nose down pitching moment upon thrust application which would act to reduce the AoA. In something like a C172, there tends to be a nose up tendency on the application of power.