You're all over the place. Coils, throttle, kill switch, neutral switch, CB500...
We've covered the throttles in two places now. A good reason to stick to one thread. Did you lift one of the slides and look at the throttle plate?
The bike will start without pulling the clutch, but it has to be in neutral. Otherwise it would lurch forward. I'm sure the CB500 is the same. (Also discussed elsewhere.)
The kill switch is simple to test. If you have 12 volts at the coil + terminals it's on and working.
You said the starter was turning the engine over, then you said it wasn't. Which is it?
The neutral light going out could be a ground problem, related to trying to start the bike with the engine ground not connected. (It should not go out when you press the starter button or when you pull the clutch in, only if you shift the transmission into gear.) Have you verified a good ground between the battery and frame? I assume you've reestablished the engine ground. Fix them first. If your grounds are flaky you'll have all sort of mysterious, confusing problems.
Coils are easy to test. Don't add new parts to the mix unless you know replacing them will be an improvement.
If you decide you need to replace the coils, search the internet for the coil-on-plug modification. Saves money and works better.
Successful troubleshooting, especially successful electrical troubleshooting, requires a logical. methodical, step by step process. Focus on one issue at a time, and don't change things until you understand what is going on.