The Talmud states that the people were tamei either from carrying Yosef's coffin, or from burying Nadav and Avihu (who died Rosh Chodesh Nissan of the second year).

Neither of these reasons would have applied on the first Pesach.

Also, even if some people had touched a corpse, the laws of tum'ah were not given yet, so they would not have actually been tamei.

The laws of tum'ah were not given until the Torah was given on Shavuos (or possibly later, depending on whether all mitzvos were given on Shavuos). Thus, even people who touched a corpse would not have been tamei and were able to bring the Korban Pesach at the correct time.

Note that in the verses about the laws of the first Korban Pesach, there is no mention that a tamei person can not bring it.

The Talmud states that the people were tamei either from carrying Yosef's coffin, or from burying Nadav and Avihu (who died Rosh Chodesh Nissan of the second year).

Neither of these reasons would apply on the first Pesach.

Also, even if some people had touched a corpse, the laws of tum'ah were not given yet, so they would not have actually been tamei.

The Talmud states that the people were tamei either from carrying Yosef's coffin, or from burying Nadav and Avihu (who died Rosh Chodesh Nissan of the second year).

Neither of these reasons would have applied on the first Pesach.

Also, even if some people had touched a corpse, the laws of tum'ah were not given yet, so they would not have actually been tamei.