A few pointers......
Wasn't there something about the switch gear on the right differing between the KH and the S3 bikes? Something in the way the kill knob works?
I don't know if you changed it or not, so I'm mentioning it.If you test the points with a voltmeter, you should read 12V or abouts with the ignition on. You can test all three by rotating it until the points open, then checking the voltage.
S3 bikes, from the experience I have with mine, will not spark right without a battery. they will spark but may only run on two cylinders. Flipping the headlight on will make the third chime in. this is usually caused by a bad or missing battery.
The coils should read 12V between the two small posts on the coil when the ignition is on and the points are open. If you leave it sit with voltage on it, the coil will get warm.
If you have 12V to the points, you can toggle them with your fingers open and closed while the key is on and watch the plug fire. You don't have toe kick the bike. It will do it the minute the points open to ground. you can test all three that way and also tell if you have the coils connected to the wrong cylinders. Just lay the plug on the head, use your finger to lift the lever on the points, and you should see spark each time the points break their ground connection.
If I remember correctly, the coils are marked + and -, make sure you have the wires connected and not cross connected.
The ground strap between the frame and motor countersprocket cover is required due to a rubber mounted engine not having a good path to ground. check with an ohm setting between the block and a good metal connection on the frame. the ohmmeter should almost zero out with a good connection.
You may want to check your ignition switch and wiring if the voltage is missing from the points when they are open with the key on. I had one of the wires corrode to a point the bike shut off while riding it when I hit a bump. The white wire, if I remember correctly, supplies the 12V to the switch.
S3 coils don't ground through the clamp as they are isolated systems. there is a ground connection on the coil itself one of the wires attaches to. (I believe black with yellow stripe).
Hope this helps