Julie Andrew's narration in One Man's Dream (and the Share A Dream Come True Parade) actually felt personal and heartfelt when reminiscing about Walt.
Iger re-recording Andrew's lines doesn't capture the same sentimentality as she did. Especially since unlike Andrews, Iger never had a personal relationship with Walt. Just like at this cute behind-the-scene photo Julie Andrews and Walt did during production of Mary Poppins.
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Heck, Iger was 15 years old by the time Walt died in December 1966.