What happens if an aircraft is only stable at higher angles of attack?
I find it hard to come up with an actual example that behaves like postulated. Normally, airplanes are most stable at low angles of attack and lose this stability at higher angles.
But there is indeed one case where angle of attack stabilizes the airplane: A swept flying wing without a vertical tail. Like the Horten designs. Those would enjoy increasing directional stability with increasing angle of attack. Your limits are a bit extreme, but a Horten IV was directionally indifferent at high speed when the typical angle of attack of the root airfoil would be between 2° and 0°. At lower speeds the induced drag would help to pull the airplane out of a sideslip as explained here.
Due to the high inertial moment around the vertical axis this behavior can be controlled by the pilot because sideslip angles will build up slowly. But it is not pleasant since it needs constant control inputs and puts an unnecessary workload on the pilot which might interfere with other tasks like navigation and collision avoidance.
Of course, this example is not quite what you wanted, since a higher sideslip will not stabilize the airplane. Only pulling up will.
If we hypothesize that an airplane behaves as you postulated, it will tend to leave the unstable region, but this might happen both in positive or negative AoA direction. An excursion into negative angles of attack is rather unpleasant (have you ever flown an outside loop?) and will probably make the pilot pull up hard. What happens next? The airplane flies through the unstable AoA region and picks up more pitch speed, only to arrive in the stable region with such a force that it overshoots the AoA limit and either stalls or, given enough flight speed, breaks up.
The Saab JAS-39 Grippen had such an unexpected overshoot at the most unfortunate of times, namely during a flight display on the occasion of the yearly Wattenfestival in Stockholm in 1993.
Any instability in the regular operating region is a bad idea and an invitation for mishaps that could be easily avoided.