Here are some simple pictures to go with my explanation:
Normal Circuit (upper lead to spark plug then ground, lower lead to kill switch then ground - ground potential is the same between the two grounds) :
View attachment 434100
Circuit open (spark plug firing/kill switch open):
View attachment 434101
Circuit closed (spark plug not firing/kill switch closed):
View attachment 434102
If there is any part of that circuit along the way that is touching ground (exposed wiring, conductable debris...etc) then you can end up with a no-spark situation. You said with the kill switch disabled (removed from the circuit), you got full spark, however when connecting it back up you got nothing again. Using new larger gauge wire you got small spark...at least that tells you that it is working, though not where you want it. Might be due to the larger gauge wire having more resistance and only letting some voltage through. Try to match the gauge wire of the saw, not sure it might be 22 gauge (just a guess here). If you have an old cell phone charger you are not using, you might be able to cannibalize the wiring from it and have a more close in size wire to work with - at least to test.