While the player in seat 1 was expecting to be skipped for the next hand, once seat 8 is eliminated, seat 7 (ok, 6) gets an unfair advantage of being the button twice. Combine this with the disadvantage of seat 1 missing a hand, allowing seat 1 to be the button is an obvious option that is supported by rule 10.
Note: the fact that seat 7 was also eliminated is irrelevant.
For the record, I used to believe that seat 1 would be skipped the next hand in the situation. Then I started to think about it, and checked the rules. Bingo, rule 10.