All parts and circuits are designed to fulfill a specific purpose. In the case of an amplifier, the design should be selected based on input, output, control function, etc. If, for instance, I wanted to use a 3 volt supply, I would use an amp that was designed to operate at 3 volts - or be easily adaptable. This eliminates the disadvantage of trying to adapt to an inherent problem.
When you say that a part or circuit has disadvantages, that usually means that it it not designed to do your intended job easily. Your statement is pretty broad and seems to assume that just because an amp has a high input Z, it is inherently ALWAYS at a disadvantage. Having no other information, I agree with you - I don't understand.
Now, if the statement was made with the understanding that the high Z amp was being considered for an application for which it is not intended, the statement could well be true.