The capacitor that bites its own tail

Hello everyone!
I take advantage of this "Project" section to explain one that is very close to my heart and that, I hope, will interest you very much.
In this project, I simply want to charge a capacitor, but I want to replace the "traditional" resistor, with the primary of a small transformer.
What idea, isn't it?
First phase:
If I pass direct current, the capacitor will charge "slowly" because the winding of the transformer will only be seen as a simple resistor.
So I chop the charging current (square wave) because I absolutely want to operate my transformer. I mainly use an Arduino and a transistor inserted into the load line. (switching)
I still see that little current passes.
And for good reason: The secondary of my transformer is not connected to anything (in the air), then, the primary behaves like a self, and strongly blocks the passage of the charging current. (filter)
So I have to put on the secondary of my transformer a load resistance (R1)
so that my secondary consumes energy and indeed, my capacitor quietly resumes its charge.
There, of course, no one wonders how this current that passes through R1 on the secondary can appear, (we can replace R1 by a small bulb, to be convinced) while the current that passes through the primary charges, AT THE SAME TIME my capacitor.
(verifiable by voltmeter, between the most of the capacitor and the mass, the amount of accumulated charges is then deduced by simple calculation)
One of the interesting properties of galvanic insulation is that it doesn't care where the primary energy comes from (as long as there is) and also doesn't care where it returns its secondary energy (as long as it can be consumed).
A well-designed transformer can have an efficiency of more than 90%.
So, followed by a diode bridge, this secondary current can therefore very well recharge an accumulator and in this case the same accumulator that was used for ... charge the capacitor. (minus inevitable losses)
Balance sheet: The energy that was used to charge the capacitor was used at the same time to recharge the accumulator, minus some inevitable losses.
And there, I see your hair bristling.
"But this mad is talking to us about efficiency!"
But admit it, you have acquiesced all my statements so far. No?
End of the first phase.
Because that's not all: I have my capacitor always charged... who is just waiting to discharge by following the same process: Hashing the discharge current, passing the current in the primary of the transformer, (which now returns to the mass) recovery of a current induced in the secondary, passage through the diode bridge, recharging the accumulator, always and again with some losses, of course.
But, you will note well, in this phase of discharge, my assembly did not consume anything at the terminals of the accumulator.
I am ready to read your thoughts and accept your help.image4000.pngimage4000.png