Is the Afikoman meant to be a substitute for Korban Pesach?

Question

I don't recall, offhand, what the purpose of the Afikoman is. Is it supposed to "represent" the Pesach sacrifice?

If so - I'm curious about a common instruction I see in many Hagaddot. It says that one should not stuff himself so much during the meal that he has no room to enjoy the afikoman and feels forced to eat it. I got that idea. However, if the afikoman is to represent Korban Pesach, I would think that one should specifically satiate himself (i.e., intentionally eat a lot so that he is full) because the korban Pesach was supposed to be eaten al hasova - after being satiated.

Answer 1

The Rishonim dispute whether Hillel would wrap the Korban Pesach together with the Matza and Maror(Rashi Pesachim 115a s.v. Shehaya and Rashbam ad. loc. s.v. Korchan) or eat the Korban Pesach separate from the Matza and Maror (Rambam Hilchot Chametz Umatza 8:6-7, Rabeinu Chananel ibid). This dispute accounts for the variations in the text of Zecher Lemikdash Kehillel that we recite before eating the Korech (see Baer Heitev475:9 and Mishna Berurah 475:21). Some versions state, "He would wrap Matza and Maror and eat them together," in accordance with the Rambam's view. Other versions state, "He would wrap Pesach, Matza, and Maror and eat them together", following the view of Rashi and the Rashbam.

According to the Rambam and Rabeinu Chananel, Hillel and Chachomim:
Both hold that the Kezayis of Pesach offering was eaten right at the end (al hasova) without Matza and Maror so the Afikomen is merely a representation of the Pesach offering and therefore should be treated like the Pesach offering to be eaten when full but have a room left. Hillel just reasons that the Matza and Maror have to be together beforehand whereas Chachamim say one can eat them separately a swell.

The opinion of Hillel (Chachamim don't argue they just say eat separately if you want)According to Rashi and the Rashbam:
The main Matza and Maror were eaten right at the end together with the kezayit piece of Pesach lamb/kid in the time of the Beis Hamikdosh, and no Matza or Maror was eaten beforehand. So nowadays the pesach is not availlable and the marror is no longer a Mitzva Deoraisa, so the Afikomen at the end is a representation of the actual Matzas Mitzva which accompanied the Pesach offering. This is why Rashi and the Rashbam say Pesachim 119b:
אין מפטירין אחר מצה אפיקומן - שצריך לאכול מצה בגמר הסעודה זכר למצה הנאכלת עם הפסח וזו היא מצה הבצועה שאנו אוכלין באחרונה לשם חובת מצה אותה שלאחר אכילה - The Afikomen should be eaten as the main Mitzva of Matza on Seder night as a rememberance to the Matza that was eaten together with the Pesach Offering at the end of the meal al hasova (i.e not being stuffed but almost full).

Answer 2

Here is a wonderful Shiur at Olamot that addresses all that you asked.

In short, it's a Machlokes why the Afikoman is eaten during the Seder, either because it is a Zecher of the Korban Pesach, or that it is part of the Chiyuv to eat Matzah, or because Chazal wanted the taste of Matzah to remain in our mouths. Over time, it seems to have become a weird hybrid of all of these Shitos.

As for not stuffing oneself, we obviously want to avoid potential Achilah Gasah issues. See there.