Why will we get the wrong answer if we use \$I_1=\frac{V_1}{Z_1}\$ to calculate \$I_1\$ directly?
You haven't drawn the circuit but how do you know that the impedance on the primary side is across the input voltage 600V? 

I=V/Z holds true if that 52-j30 Ohm impedance were in parallel to the input souce. And if that is the case, the current through that impedance, still would not be the primary current going into the primary side of the transfomer.

This is what you're actually doing:







simulate this circuit &ndash; Schematic created using CircuitLab

\$I_1\$ is the current through the 50-j30 impedance, but is not the one that actually flows into the primary side.

Maybe, just maybe this is what you really have:







simulate this circuit

And in this case, all the primary current flows through the 50-j30 impedance, and it makes sense to find that current with the \$I_P\cdot N_1=I_S\cdot N_2\$ relationship since you already have all you need.