Why does my laptop adapter output nominal voltage with a broken capacitor?
Yes, the voltage across a capacitor with a load and without a load is vastly different.

What you are seeing there is the rectified AC signal, which is very very rough, just about making the power supply work.  The capacitor filters and smooths that rectified AC signal:

When the capacitor is faulty, with no load, it can store a little bit of charge and the green line remains fairly smooth.  As soon as you apply a load the capacitor is no longer able to hold enough charge, and so the green line more closely follows the blue line.  Because of that there then isn't enough power to run the power supply.