Average amplitude and DC component
Two examples:

Suppose you send a signal with a DC component to a loudspeaker. The cone of the loudspeaker wouldn't then move around its center position, as it is designed to do, but be pushed out constantly by the DC component.
It will move around that pushed-out position to play music, but not moving around its center position will give distortion. The average amplitude should be zero, so this is a case where you don't want a DC component in your signal.

Suppose you want to sample a signal with an average amplitude of zero. Unfortunately, most ADCs can't sample negative voltages, so you will have to lift the whole signal far enough for it to stay above 0V all the time.
This means you will have to add a DC offset, so that the ADC can sample the signal. This is a case where you do want a signal with a DC component.

There are no general pros and cons to having a DC offset; it all depends on the circuit handling the signal and the purpose of the signal: sometimes you want or need a DC component, sometimes you don't.
The terminology is just there to describe the signal; it is not a value judgment.