The exciter coil windings should produce a voltage much greater than 12 volts, but . . . ALTERNATING CURRENT.
Do you get a pulse from the pickup coil?
Rather than cut wires to the kill switch, I'd recommend checking the continuity of the wires indicated by the ignition switch matrix on the KLR650 wiring diagram
(Google bold print to view a copy; KLR600 ignition essentially identical to that of the Generation 1 KLR650, AFAIK
So . . . with adequate alternating current from the exciter coils to the CDI, a pulse from the pickup coil to the CDI, and operational CDI and ignition coil, spark will occur. The kill mechanism is grounding the CDI power lead, by either the kill switch or the OFF ignition switch position; a continuity check should tell you whether an incorrect grounding of this vital voltage is occurring.
The pickup coil lead to the CDI passes perilously close to the countershaft sprocket; continuity/short circuit may be an issue here.
I think a wiring diagram and a multimeter are your friends in this case.
(Comments from my experience/opinion; corrections/clarifications welcomed from the readership!)
This link leads to a reasonable CDI tutorial: https://4strokes.com/viewtopic.php?f=58&t=7