I would say the main issue is the inverted logic in comparison with a series regulator: in a parallel regulation scheme, you need to burn all the power you don't need, both in the regulation element and in the series resistor. On full power, you reach the same efficiency as the equivalent series regulator, but in all other cases you are wasting more heat.
Given the fact that power supplies are seldom used continuously at full power, you would end up with a nice stove for winter times. Heat causes then quicker aging of the system.
Output impedance in a power supply is basically never considered by the traditional designer (plenty of filtering on the DUT); in case of need he/she will buy a 4-quadrant supply.
Short circuit protection as well: old problem, solved in many ways since the early days of electronics. Basically every power supply is short circuit tolerant nowadays.