Your power supply is a 0-5A supply, so it is current limiting if it sees too low a resistance (too high current). You cannot raise the voltage because of this, the power supply is limiting the voltage so the current does not exceed 5A (5.16A in your case).
If the power supply allowed you to raise the voltage, the current would also increase, which is not possible with the power supply due to the current limiting.
Even if the coil is oscillating, the current is too high.
Of course, if you have something wired wrong, or a bad component, that could be shorting out your power supply as well.
Notice in the video that he is using a battery which can supply high currents, probably like 20A for short periods of time. Finally, you definitely need a heftyheatsink for the transistor. Without one, you can blow that transistor within seconds due to high heat buildup