Before Cantor, there was a great deal of vacillation about which
operators to use for talking about sets or collections (or Menge).
Cantor wasn't the first to suggest an isIn or elementOf operator,
but he used it -- as you observed -- as the primitive.
I'm not sure what you have in mind, John. Set theory didn't exist in any systematic form before Cantor and sets per se weren't objects of mathematical interest; at most they showed up in the analysis of logic, as you note in the cases of Boole and Peirce. Cantor's own theory of sets arose out of his work on transfinite arithmetic, not logic. Granted, the logical and Cantorian (what Gödel called the "combinatorial") conceptions of set weren't clearly distinguished until after the discovery of the paradoxes, which arguably originate in a failure clearly to distinguish the two. But it is noteworthy that Cantor himself (popular myths to the contrary) never saw any paradoxes in set theory as he'd developed it, as he saw that certain principles allied to the "logical" conception (notably Comprehension axioms) did not hold for his conception.
But pedagogically, it helps to start with Boolean algebra and show
how many other theories can be developed as specializations of it.
I'm not at all sure that's the best way to study set theory; indeed, I fear it encourages the confusion between the two conceptions of set noted above.