Could be OP-AMP in current, or voltage control, or bad output transistor, which usually cause that issue. Without a schematic and measurements it a guess at best. Do you have a scope, or meter around to troubleshoot it? As you need to check taps off transformer, output transistors, relay section, big power resistors and OP-AMPS input and output. as for CV, you monitor the output voltage and feed this back into the inverting input of the op-amp driving the output, for CC, you monitor the voltage across a current sense resistor and feed this back into the same inverting input of the op-amp, is usually how it works as the circuit will automatically switch into CV or CV depending on which threshold is reached first. If current op-amp isn't working correctly, you won't have current control and voltage will drop at random, or no output at all.
First I would tap the board and make sure no cold solder joints and look for any obvious signs of damage components. Next step to take is to check transformer taps, relay section, big resistors, output transistors section and make sure no issues as shorted, or open transistors bad diodes, or resistors, ESR on capacitors and also getting correct voltages with the power section first and work your way back. Also check for excessive ripple. When you get to OP-AMP's assuming you check everything else. One trick is to remove the current OP-AMP and that will keep power supply from going into constant current mode and if you get stable voltage then current OP-AMP, or potentiometer is bad. Just make sure you don't draw more current then the PSU rated for, as you will blow the outputs, as you will have no current protection till OP-AMP is replaced.
To set current limit set power supply voltage to around 5 volts DC and short the + and - together to set current limit, then turn current pot to when the display read 25ma, so you don't blow LED, or put to little current in.
I watch the video, one thing I would like you to try is to short the + and - set voltage to 5 volts and turn the current knob and then tell us if it has a affect on current display, if current goes up, or down when you turn the knob? Plus if you can turn current knob down all the way on both coarse and fine and if CC light come on? One warning if current shoot up with no control pull cable, or turn off power supply. If OP-AMP is bad will cause lost of current control and issue with voltage dropping and output cutting off and on.