OK let me take a stab at this...
Yes the cap smooths the full-wave bridge-rectified DC and also raises the average voltage to the level of the top of the peaks. Bad news is when you go to start the arc you get a big nasty snap :-( , not good...
Having just an inductor choke coil is good for starting the arc and for maintaining the arc.
The coil limits the inrush pulsating DC current when starting the arc.
The coil also maintains the pulsating DC arc nicely once started.
All I can say is if you insist on a big fat cap, then better have a "large" choke coil to limit the inrush current when starting the arc.