How can I trigger an AC output ON when an AC input is OFF?
OK, just as an academic exercise, here's one way to achieve what you want. The transistor circuit inhibits passing current to the motor if there is also voltage on the control input (R1). Otherwise, the pass transistor behaves as a \$\frac{100 \Omega}{\beta}\$ resistor. Two bridge rectifiers allow it to work on both halves of the AC cycle.







simulate this circuit &ndash; Schematic created using CircuitLab

Select a transistor for Q2 that can handle the motor current.

Equivalent circuit for positive half-cycle:







simulate this circuit

Equivalent circuit for negative half-cycle:







simulate this circuit