In video above, they use same system as you suggested. Only they have used isolation transformer.
In my opinion that schematic have high ripple and relatively large capacitance on output. So short circuit energy is very large... That means current control would have huge overshoot, until capacitor is discharged.
We don't know spec for PSU, that maybe acceptable, but not very nice solution in my opinion.
I definitely would go with minimal output capacitance, overshoot and ripple.
All that you can get from solution that I suggested.