Don't see how it's possible to feeding anything from the starter solenoid to the CDI directly unless you have a wire or two shorted out or a miswired connection.
However, to alleviate your concern over the solenoid. Remove it from the system.
To start the engine run a heavy gauge wire from the positive terminal of your battery to your starter TEMPORARILY TOUCHING THE STARTER LUG. Once started RELEASE the wire. Check your kill switch now and see if its working... IF its still not working your solenoid should not be an issue with what you are describing.
What typically feeds voltage to the CDI should be coming from the stator to the #6 position wire on the CDI unless you have a setup I have not ran across.
For instance, the wire on my #6 position on the CDI is a black wire with a red stripe fed directly from the stator.
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