…[I]n accordance with the opinion of Rabbi Natan, who said that failure to engage in eating the Paschal lamb does not preclude one from fulfilling one’s obligation to bring the offering, as the eating is a separate mitzva.
The Gemara searches for a source for this opinion of Rabbi Natan. One suggestion is the statement of Rabbi Natan that the whole Jewish nation can fulfill their mitzva with a single korban, something that he derives from the passage (Shmot 12:6) which teaches that “the entire Jewish people should slaughter it in the afternoon.” While the Jerusalem Talmud accepts that as the source for Rabbi Natan’s opinion, the Babylonian Talmud suggests an alternative source, as well.
The Tosefta teaches that if two groups of people choose the same animal to be their korban, according to the Tanna Kamma, the first group eats a ka-zayit – an olive-size piece of the sacrifice – and fulfills their obligation; the second group, who do not have an olive-size piece of the korban, will have to bring another sacrifice on Pesah sheni. Rabbi Natan says that neither group will have to bring a sacrifice on Pesah sheni; once the blood was sprinkled on the altar, all parties to the korban have fulfilled their obligation. According to Rabbi Natan, the passage (Shmot 12:4) obligating the people to take into account the amount that each person will eat when joining a group for the korban Pesah simply means that the person who joins the group must theoretically be able to eat the sacrifice. It does not mean that the Pesah must be eaten.
Without question, even Rabbi Natan agrees that every Jewish person should eat a portion of the korban Pesah, and someone who did not eat from the sacrifice would miss out on the opportunity to fulfill this mitzva. The point of disagreement is whether bringing the korban is sufficient after the fact, even if it was not eaten, or if someone who does not eat has missed out on the mitzva entirely and will need to bring a second korban when the opportunity arises on Pesah sheni.