[1] I prefer to translate tumah as unprepared as opposed to its usual translation of impure. What tumah signifies is that one is not in the proper state of mind to come to the Temple and offer sacrifices. Either one was unfocused allowing themselves to come in contact with tumah, or they purposely came into contact with tumah by going to a funeral in which case they need time to recover to enter the Temple with the requisite feeling of joy. As the Mishna teaches there are exceptions to the general prohibition of entering the Temple in a state of tumah.
[2] It is because of this omission that the possibility exists of bringing the korban pesach even without a Temple. “I will lay your cites in wasted and destroy your Temples and I will not savor your satisfying aromas” (Vayikra 26:31). G-d does not want our (meaningless) sacrifices - they give off a “foul odor”. The korban pesach focused on our homes may be offered even if we have blunted the smell of other korbanot.