The aircraft will spin, and I have done so a few times for flight testing, along with the 181 and the Arrow III and IV. The problem as I understood it (way back when) was two fold. Firstly, when the aircraft is really 'wound up' the 'break out' forces can be 'misinterperated' and while you might think you have full control deflection, in fact the controls come up against a small aerodynamic 'stop' and a greater control force is then required to 'break out' of the spin. This was particularly so with a rear limit C of G. It was thought that some pilots might not recover because they would be lead to believe they had applied full recovery action when in fact they had not.
It had not escaped the notice of Piper, at the time, that these aircraft were being loaded without any regard for the C of G or calculations to confirm the aircraft was within the C of G limits. ie. four thumping great chaps and a big old dinghy and baggage in the back. As the aircraft was not intended for aerobatics, the second factor was thus one of 'Product liability', so it was easier to just prohibit this maneauvre. At the time Cessna had just been taken to court under the product liability issue and lost to a man who ran out of fuel and crashed. His argument was that nowhere in the flight manual (aircraft handbook) did it actually say that you had to put fuel in it!!!