Why does the primary coil of a transformer not short circuit when AC voltage is applied?

Because it is an inductor - inductors oppose change in current; so every time the current changes (continually in AC) the energy is transferred and becomes a changing magnetic field - and according to various laws these magnetic fields oppose the current and limit it - so doesn't act as a short.

DC sees the resistance of the coil only (low R) but not the inductance, as there is no change with DC - so DC sees it as a short........