Saharaland

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Notice that the round name, flavor text, and puzzle names only use the vowel A. This suggests we should be suspicious of any non-A vowel in the answers. In fact, there is exactly one E, I, O, or U in each of these answers (and no Y, to avoid confusion). This non-A vowel can be replaced by a unique consonant to form a new word:

PuzzleAnswerNew WordNew Letter
[Pop Music]ARIANAARCANAC
[Zagat]RATERATSS
[Island]SAMOASAMBAB
[Disney]BEASTBLASTL
[Hair]BRAIDBRANDN
[Cars]ACURAACCRAC
[Fighting]KARATEKARATSS
[Google]SEARCHSTARCHT
[Shakespeare]MANTUAMANTRAR

Reading the new letters yields the nonsense string CSBLNCSTR. However, if we add in the vowel A, we can create the clue phrase CASABLANCA STAR, which suggests the answer HUMPHREY BOGART.