Based on the passage that teaches the halakha of Pesah sheni (Bamidbar 9:10-11), the replacement sacrifice brought by an individual who could not bring the korban Pesah at the proper time because he was tameh (ritually defiled), the Sages learn that such a rule applies only to an individual. If the entire Jewish community had become tameh, then the korban Pesah would be sacrificed and eaten at its normal time, even though they were all ritually defiled.
The Mishna (76b) teaches that there are other sacrifices that, like the korban Pesah, would be brought at their appropriate times if the entire community was tameh; however they differ from the korban Pesah in that they would not be eaten. These sacrifices include:
Korban ha-omer, which is brought on the second day of Pesah. After that date the new grain is permitted.
Shtei ha-lehem, which is brought on Shavu’ot (Vayikra 23:17). After that date the new grain can be used in the Temple service.
Lehem ha-panim, which are placed on the shulhan in the Temple every Shabbat (Vayikra 24:5-9).
Zivhei shalmei tzibbur, which are brought with the Shtei ha-lehem on Shavu’ot (Vayikra 23:19).
Se’irei Roshei Hodashim, which are brought as a sin-offering on the New Moon (Bamidbar 28:15).
The Gemara on our daf discusses the source for these rules: the word mo’ed – a specific time – a term that the Torah uses with regard to each one of these sacrifices. The Gemara points out that we cannot learn these rules from one another because each one has its own tzad hamur – a perspective from which it is more stringent than the others. For example, the Gemara says that the korban ha-omer and the shtei ha-lehem come le-hatir – to permit – as opposed to the public offerings that come le-khaper – to atone.
Here we see the Gemara balancing between two values. Certainly the power that a sacrifice has to offer atonement would seem to make it a uniquely dominant korban. On the other hand, the force of a sacrifice to permit something that the Torah rules as dependent on it would seem to make it a power to be reckoned with. The Gemara thus shows that each of these has their own element of uniqueness which does not allow it to be learned from the other.
The korban Pesah is distinctive in that it is brought for the sole purpose of eating it, which allows it not only to be sacrificed when the community is tameh, but to be eaten, as well.