Is there a disadvantage to not having a power switch on a logic circuit?
Are there any major drawbacks or disadvantages to not having an on/off
  switch?


Depends on what you think is 'major'. 

If the circuit is mains powered then it should be designed to start up reliably even if 'turned on' by simply plugging it in. This may require a more sophisticated reset circuit to ensure that sequential logic is properly initialized when the power supply voltage rises slowly or erratically.

Another drawback that might be considered 'major' is if the device is battery powered, since you would have to disconnect the battery to turn it off. Some mains powered clocks use a 9V battery to keep the timing going during power cuts. If you unplug the clock for a long time and forget to disconnect the battery it will go flat.