Vayikra 22:28 (Emor) – One cannot sacrifice a mother and her young on the same day

Vayikra 22:28 records that an animal and its mother (also father? see Rambam, Laws of Slaughter 12:11) cannot be sacrificed on the same day. This law continues the previous law, 22:27, as after one waited eight days to offer the newborn animal, one cannot offer the mother as a sacrifice on the same day. This law is not just by sacrifices, as even by “regular” meat, one cannot kill an animal and its mother on the same day, see Rambam, Laws of Slaughter 12:1,2. 

What is the rationale for this prohibition? A popular answer is that the law is for humanitarian reasons just the question is whose feelings are being taken into consideration. According to the Rambam (Moreh 3:48) the concern is for the feelings of the animals, while the Ramban (on Devarim 22:6) and Luzzatto (on 22:28) suggest it is the feelings of the person killing the animal since the goal is to “cultivate human pity,” see N. Leibowitz, pp.208-210. However, Milgrom (2000, p. 1884) rejects this approach since is it any less cruel to kill the animals on two consecutive days?

My guess is that the requirement to kill the animals on different days is because this law forces one to separate the animal from its mother (and father?). The need to separate is one of the principles of numerous laws, such as kashrut, and this is one more example of the application of the principle of separation. (Is this also the principle in sending away the mother bird in Devarim 22:6,7?)