I have the following circuit built from the 'choppy' pwm controller kit.
However upon testing it, it only ranges the voltage from 10-12V.
How would one analyze the circuit shown to confirm this is the correct output?
Your MOSFET has an internal output capacitance (Coss) in the range of 500 pF. You can think of it like a parallel capacitor between drain an source. If the FET is conducting this capacitor is discharged. If it turns off, this capacitor will be charged via the load resistance.
With a motor as load this charge time is very short and will not affect the voltage reading of the multimeter, but without a load the charge time is much longer, because the charge current through the 10 Mohm internal resistance of the multimeter is in µA range only.
The voltage across the multimeter drops during the charge phase (FET off) but the next conducting period starts long before the charge can complete. As a result the average voltage across the multimeter cannot fall below 10 V.
With a LED as load you probably add some more capacitance and the forward voltage of the LEDs will also not charge this capacitance. You can think of the LED chain as a Z-diode not conducting below Ufwd of the LEDs.
A proper voltage integral reading with a multimeter needs a pure resistive load here with a current of at least 10 mA.