At least one (big IMHO) consideration is modulation linearity.
From the "Radio Handbook", 1940 edition, page 160:
Class C Plate Modulation. The characteristic of a class C amplifier which makes it linear with respect to changes in plate voltage is that which allows such an amplifier to be plate modulated for radio telephony. Through the use of higher bias than is required for a c.w. class C amplifier and greater excitation, the linearity of such an amplifier may be extended from zero plate voltage to twice the normal value. The output power of a class C amplifier adjusted for plate modulation varies with the square of the plate voltage. Since this is the same condition that would take place if a resistor equal to the voltage on the amplifier divided by its plate current were substituted for the amplifier, it is said the stage presents a resistive load to the modulator.