Why doesn't my bench PSU read current on a breadboard?
These cheap PSU usually have current / voltage reading but they are not by any mean accurate. They are more there to give a rough idea of what current is flowing.

An LED typically draws 10-20mA and your PSU, from the datasheet has a resolution of 0.01A (10mA).

Also the resolution is "what is displayed" and does not represent the actual accuracy of the device, which is probably on the range of 5 to 10% (EDIT in fact 33% measured by OP). At this price, is certainly not calibrated and you can see from the picture the current sense seems to actually be 2 copper wires used as shunt, which shunt value will change with soldering and small change in the wire thickness and length.

To answer your question, the current you are drawing is simply too small for the resolution and accuracy of this device, it probably has some offset, calibration drift, temperature dependencies and everything you can think off.

Try to hookup a 100ohm or less resistor and see if it displays anything.