I'm not sure about modern implementations, but it's not actually required to remove the old index at the end. It's perfectly legitimate (and somewhat safer, though the index should be reconstructable via a linear scan of record headers within the archive) to just append, and the old indexes in the middle of the archive will be ignored.

But you had multiple backups on these tapes. If you rewrite the index how do you restore from a certain day?

Your end-of-stream index would remain in place with a backup number / id.

Your entire index would be the logical sum of all such indices. Think of the end of stream index as a write ahead log of the index.