Why can't model aircraft be used to discover spin and stall behavior of manned aircraft designs?
Who says scale models cannot be used for spin modeling?

The key information is that the Reynolds number does not affect the behavior of separated airflow as much, so you do not need to increase speed to compensate for the smaller dimensions. Only when the stall behavior is studied would the Reynolds number matter, but here also meaningful results are possible at lower speeds. 

Models are routinely used in spin tunnels. Here both the speed and the dimensions are smaller, but the results can be transferred to the original aircraft. There are two sorts of spin tunnels:


Free tunnels with an open, divergent section in which the model has to be thrown by a skilled operator so it will settle in a spin within the upwards flowing air stream. The section is divergent, so the speed changes with height and allows the model to find a matching speed.




Picture from spin tests on a Grumman E-2 model in the NACA 20 foot spin tunnel (picture source)


Closed tunnels in which the model is mounted on a sting. The sting is connected to a rotating balance, which can be set at different roll and yaw rates. The resulting matrix of coefficients over angle of attack, angle of sideslip and over the three rotation speeds is fed into a computer model which computes the equilibrium points.


The models used in free tunnels must be scaled both geometrically and inertially, so their mass distribution matches that of the original. If the spin tunnel uses a rotating balance, not even the dynamic scaling of masses and inertias is needed, and a regular tunnel model can be used. However, if the resulting spin is of an oscillating nature, the free spin tunnel is at an advantage, because this becomes readily apparent in the test. In a closed tunnel you just get two equilibrium points and must make the connection yourself.

A third way are free-flight tests of models, but they are much more expensive and allow less observations than wind tunnels.

The stall characteristic is a little harder to predict using models, but again you can draw conclusions from what can be seen in the tunnel. To arrive at the maximum lift coefficient starting from the measured value at Re = 1,000,000, you may use this scaling: $$\Delta c_{L_{max}} = \frac{log_{10}\left(Re^{\frac{1}{6}}\right)}{3.5}$$

The plot below shows the difference in lift coefficient between a small model and a full-scale airplane. The biggest differences are around stall, and at the high angles of attack seen in spins both show fairly similar behavior.



Lift coefficient over angle of attack for model and full-scale aircraft, taken from Joseph Chambers' monograph on testing with models (Modeling flight : the role of dynamically scaled free-flight models in support of
NASA’s aerospace programs).

For small aircraft, the relative expense of a tunnel test is normally inadequate. Instead, the real thing is used. Since the speeds involved are small, spin tests are carried out with a spin chute or a releasable mass at the tail of the aircraft. On modern designs, even a full-aircraft ballistic parachute system can be employed to prevent a crash if the spin test ends in an unrecoverable situation.



Spin chute installation on a Columbia 400 (picture source)