The governor, the Tirshata, is apparently Nehemiah himself as in Nehemiah 10:1 (KJV):
Now those that sealed were, Nehemiah, the Tirshatha, the son of Hachaliah, and Zidkijah
The words "therefore were they, as polluted" do not appear in the MT. These words are an incorrect interpolation by the translator. There is no concept of "pollution" regarding family lineage in the OT.
The MT uses the word ויגאלו meaning that they were "exiled" from the (active) priesthood, not that they were "removed".
The translation "stand up" is too short and too literal to be be comprehensible in English. The meaning is "until we will have a high priest and the Urim and Thumim, and the high priest can stand in the Temple and ask God through the Urim and the Thumim if these families are really of Aaronic descent".
The problem is this. Families claiming to be of priestly descent return from the Babylonian exile, but they can't find proof of their Aaronic lineage. Apparently their claim is nonetheless strong, probably backed up by witnesses.
Nehemiah has to decide whether these people will be allowed to serve as priests at all, serve in some limited capacity, or be accepted completely. He takes this responsibility most seriously.
He decides that he cannot allow these families to eat from the קודשי קדשים, the holiest of the sacrifices, but in verse 65 he implicitly leaves open the possibility that they can eat from the regular sacrifices. He leaves open the possibility of a full return to priesthood when the Temple is re-established and the Urim and Thumim are recovered.
We do not know if these families were ever fully re-instated. They probably were not, as the Urim and Thumim were never recovered and (according to most views) were missing in the Second Temple and later generations of religious authorities were less inclined to take lenient views on these matters.