Great video.... ok, you guys know I aint no electronics wunderkind but let me take a stab at the reason without using a schematic.

L1and L2 connects at the burner switches. When a switch is turned on, L2 then supplies 120 voltage to the indicator turning it on, despite the burning not actually beingon since no voltage from L1.

Normally L1 supplies voltage to the board but since L1 is dead, no power at the board until a burner switch is turned on causing voltage from L2 to be sent to the indicator and back along the neutral line of the indicator to the board thus L2 instead of L1 is supplying the board with 120v as long as a burner switch is on,

Do I at least get a passing grade for trying?

I am pretty sure Durham has this right.  I had a similar issue a while back with some bad house wiring -- L2 had 240 volts from L2 to Neutral, L1 had 0 Volts from L1 to N. Customer wanted money back because "this stove isn't working", but alas, they had to  call their builders and demand a rewiring!

My oddest finding so far -- an incinerated mouse corpse on L1. Mouse apparently had his last snack on black. 

Had one where the rats ate through most of the wire harness. I was there the same time the exterminator was checking the traps in the basement. Told the customer what I saw. He was in disbelief. Said that the rats were only in the basement. Lol