minimum load current vs stability
In the design you show, the gm of the output stage will be dependent upon the load current and so will affect the loop gain and potentially stability. Providing a minimum load will reduce the variation in gm and allow a better compromise in the feedback compensation.

Why do you have a capacitor to ground at the output of the amplifier - that will tend to make stability worse by introducing a pole in the transfer function.

Many voltage regulators have minimum load constraints for a different reason. Three pin regulators such as LM317 intentionally divert most of their internal operating current to the output in order to keep the current into the adjust pin constant. This means that if the output current falls below the operating current of the internal circuitry LDO will lose control and the output voltage will rise.