Depends on how this transmigration works
1-It happens at birth.
2-It can happen at any-point in the vessel's life and depends on their Ideology.
What Hagoromo said here suggests it might be #2
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If it was Situation #2...Itachi wasn't suited because he was a closet pacifist and shunned the Uchiha curse of hatred.
Obito would've been a good choice...but then again, he never possessed double Mangekyu.
Sasuke had the right ideology and access to his full ability.