With the database in recovering state, would all of the log backups have been needed to allow SQL to roll back the index?
No. Since the transaction (the index rebuild) was still active, all the log records needed for the recovery would be in the transaction log, log records cannot be removed if they could still be needed for database recovery
From the lack of free space in the database I don't think that it was building a new index in parallel, so I don't see where else the data for the rollback would be stored.
It was building in parallel, that's how SQL works with index rebuilds, but the rollback data would be in the transaction log, that's the only place it can be.
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: Discussions on DB performance with occasional diversions into recoverability
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