To my surprise, the RMS current through the generator is 585mA... almost twice as much as what I expected.
I just dropped in to the thread. The surprizing point for me is, that you got a lot of verbose explanations and suggestions, but nobody mentioned the simple fact, that the RMS input current will be considerably higher than the output current, and that it of course depends on the capacitor size.
I guess, that you are able to understand the effect by analyzing the input current waveform, applying the quadratic RMS calculation rule.
The simulation however doesn't exactly apply to a real transformer, because you have to consider it's ohmic (winding resistances) and reactive (leak inductance) output impedance. It will result in a lower RMS current but also a reduced output voltage. The unloaded output voltage of a small transformer is considerably higher than the assumed value. If designed correctly, it will supply the nominal voltage and current to a resistive load connected directly to the transformer.
For a 6V 1.9 VA transformer, I determined these parameters:
open circuit voltage 8.6V
output resistance 7.7 Ohm (uk,r = 40%)
output reactance 5.4 Ohm (uk,x = 28 %)
Using uk relative impedance voltage numbers, you can easily scale the parameters to a different transformer. If you know the values of larger transformers, you'll notice, that small transformers have a considerably higher impedance voltage, respectively voltage drop at nominal load.
P.S.: uk is actually the German formula symbol for impedance voltage or "Kurschlussspannung". The usual English symbols are Req and Xeq respectively.