I remember seeing that video a while back. Yes, you should be able to wire it just like that.
If there is any way for the metal to be touched, then for absolute safety the metal should be grounded. But if the power supply (with the AC connections) is in an insulated enclosure, with only the 24V output coming out, then it’s safe even without it. (There is, however, a chance the metal may feel “tingly” if it’s not grounded. This is due to the EMI filter capacitors, which allow 1/2 of mains voltage to appear on the DC output, albeit at absolutely microscopic currents. This is normal and doesn’t represent a malfunction or design flaw. Grounding it eliminates it though.)