לנפש לא יטמא בעמיו (אמור כא:א)

Rashi says: “בעמיו – יצא מת מצוה” – a Kohen is allowed to contaminate himself with a dead body when it is considered a mitzvah.

If “כל העדה כולם קדושים” – all the Yidden are considered holy, then the impurity of a dead person should apply to all Yidden, not just Kohanim.  If so, then every Yid would be regarded as a מת מצוה, so therefor there have to be Yidden that can become impure by tending to the dead, and thus the Kohanim shall remain pure, enabling them to sacrifice the Karbonos.