After reading the posted references, here is what I gleaned:
In this application, the ferro-resonant transformer is not used because it acts as a voltage regulator; rather it is used to regulate charging current, making the charger act more like a constant-current source, rather than a constant-voltage source. If you use a well-regulated voltage source to charge a lead-acid battery bank, the problem is that the initial charging current is huge (and it blows up the power supply). In the Lester charger, they are relying on transformer core saturation to keep that initial charging current down.
The core saturation also has another benefit; it squares the waveform that comes out of the full-wave rectifier such that the current flow is more like filtered DC. There is no filter capacitor in these chargers, but unlike a regular transformer-rectifier combo that would deliver a series of short high amplitude current pulses to the battery, the ferro-resonant charger delivers wider, flatter pulses, closer to pure DC.
Now here is something I haven't yet figured out. Talking to an old ham buddy about it, he suggests that the resonant circuit is tuned to the third harmonic of the power line frequency (180Hz). The third harmonic added to the fundamental makes a waveform which looks more like a square wave (by filling in the corners, knocking down the peak in the middle). This action occurs after the charging current has dropped no longer causing the core to saturate. The Q of the resonant circuit increases as the core becomes more linear (non-saturated), boosting the output voltage to finalize the charging cycle. Is this true?
In the charger I'm trying to fix, the capacitor has changed value significantly, preventing that freewheeling voltage boost at the end. I'm ordering a replacement, and will report back after I put it in...