2:62a These men searched for their family records,

The people ⌊from these clans/families⌋ searched for records ⌊that proved⌋ their descent/ancestry,
-or-
These people looked for writings with a list of the names of each generation of their ancestors,

2:62b but they could not find them

but they were not able to find such records.
-or-
but they could not find proof in writing ⌊that they were descended from sacrificers⌋.

2:62c and so were excluded from the priesthood as unclean.

As a result, they were excluded from ⌊the possibility of⌋ serving as priests because they were ⌊considered⌋ ritually impure ⌊according to the law of Moses⌋.
-or-
Because ⌊they did not have proof,⌋ this made them polluted, ⌊and the leaders⌋ would not allow/permit them to serve ⌊Yahweh⌋ as sacrificers.

2:61–62 (reordered)

Also among the priests, there were some clans who searched for records of their ancestry, but they could not find any. Because of this, they were excluded from the list of true priests and were not accepted to do the work of priests. These were the clans of Hobaiah, Hakkoz, and Barzillai. This Barzillai married a daughter of Barzillai who came from Gilead and then took the name of his father-in-law.
-or-
Also among the priests, the descendants of Hobaiah, Hakkoz, and Barzillai searched for records of their ancestry, but they could not find any. This Barzillai had married a daughter of Barzillai, a man from Gilead, and took her family name. Because these clans could not prove their ancestry, they were considered impure and excluded from the list of true priests.

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