Yes, and if you think about this it is the only option that makes sense for live use. EXAMPLE: If you deliberately delay the sound to certain outputs with a delay plugin - you want to be absolutely exact. This is used frequently with larger halls where you have to delay speakers in the middle of the room so that the front+mid speakers produce a consistent sound for the back of the room.
How would a delay that uses say 1/8th of your beat as the delay value be properly aligned with your performance in this case if it could be playing its audio delayed again by some random value?
Yes. Consider a delay that plays just the delay (wet is set to 100%). If you place 2 delays that delay the sound by 100ms - you will hear the first sound after 2x100ms