What happens when an alphabet is not listed in a Turing Machine?
The behavior of the Turing machine is not defined in this case.  So, the mathematical formalism does not predict or require anything about how the machine will behave in such a circumstance.

Since a Turing machine is only a mathematical construct anyway, it's perfectly reasonable to impose such a limitation and only think about what happens when you run it on a tape that follows all the rules.

Put another way: If it's possible for the symbol b to occur in the tape, then it's your job to make sure that b is included in the alphabet.