Why is it important that the core of a transformer pass through both coils?

If it doesn't, you don't have a transformer. The core is where the magnetic flux will pass, which induces voltage on the secondary (voltage applied to the primary winding induces a magnetic flux in the core, which induces a voltage on the secondary winding); If the core is not passed through one coil, it will not induce a voltage in that winding.

Leakage flux outside the core can result in some inductive coupling, but the job of the core is to couple the primary winding to the secondary winding.