Current protection is not needed.
Wrong it is always needed.
In this case you say:-
The maximum output current is going to be in the single digit milliamps,
So you are saying that the current is limited by the limited amount of energy transformed by the coils. But what is this? How are you calculating it and are you in control of it?
If the voltage collapses when 20mA of current is drawn then how do you know if more than one LED will work when you connect them in series?
there's not enough wattage going into the LEDs to blow then up.
Apart from the fact that you don't have "wattage" going into anything can you say how this figure is derived.
All in all it is a very hairy circuit that only appears to function due to secondary effects that are not under your direct control. It is a very bad design as a result.
I have one running that has 4 LEDs attached and has been running for over 72 hours straight off
Functionality alone does not a prove a design. Have you measured the light output over that time to see if the life of the LED has been drastically reduced?