What's the minimum electrical requirements to ac-ac conversion
An inductor with inductance \$L\$, operating at frequency \$f\$, has an impedance \$Z = 2 \pi j f L\$. For \$\mu\$H inductors at 50 Hz, the impedance would be in the m\$\Omega\$ range. But your power supply can't source enough current to maintain 12 V over such an inductor. Try measuring the voltage on the power supply side when the circuit is running - I bet it is close to 0 V.

To get the voltage step-down/up you are expecting, you therefore need the inductance of the primary to be much larger than \$ V_\mathrm{PSU}/(2 \pi f I_\mathrm{PSU})\$, in your case approximately 100 mH.