Value of the capacitor required for smoothing
The symptom you see is probably due to the larger capacitor having higher leakage.  Your overall power input is so low that the capacitor leakage eats up a significant fraction of it.  The peaks of the input waveform contain so little energy (are at a high enough impedance) that they are clipped against the capacitor voltage without changing that capacitor voltage much.

The capacitor is doing its job of averaging out the voltage, but comes with a unwanted resistor across it internally.  Unfortunately, affordable large capacitors are electrolytic, which have high leakage relative to other types.