Looking at the diagram, it's not complicated.
The magneto has two windings (and a permanent magnet not shown). The yellow/black pair drives the accessories and the white/red pair drives the ignition. They appear to be electrically equivalent, though their values in turn count are not shown, so that may not be the case. Each has a connector you can separate for testing.
Pull the connector, attach a resistor across so the winding isn't totally open circuit, and attach an analog voltmeter across the leads. Choose the highest voltage range first, then step down until you see the needle move, though I expect the 15-20V range will end up being the one you want. Pedal to spin the engine and see if there's voltage appearing across the first winding, then switch to the other and perform the same check. If they're both alive, I doubt there's anything at all wrong with the magneto. Conceivably some windings could be shorted and still let the magneto perform with degradation, but it's very unlikely.
(Your post didn't say if you had tested both coils of the magneto, nor if they appear equal.)
(Ideally you could use a scope to see the pulse from the CDI unit to the coil primary. Knowing there's something coming across the primary as you said suggests the CDI unit is likely okay.)
If both halves of the magneto seem okay, then it's 1) wiring 2) CDI unit 3) coil.
Take the coil out of circuit, put a resistor across that lead to ground, measure across with an analog voltmeter. If it pops as you spin the motor, there's a pulse to the coil primary. If not, the CDI unit is bad.
(You did say there was something coming from the CDI output. I'd prefer to see the pulse on a scope, but we make do with the tools at hand, so "something" is probably the best we have to work from.)
If there's a pulse to the primary but insufficient voltage at the secondary (do not try to measure that with a voltmeter, it's too hot), then figure it's the coil. Try replacing the coil. If that fails, then go through the wiring diagram step by step looking for a short to ground or open somewhere. Undoing all the lights first may simplify your task.
It's just a matter of eliminating possibilities step by step, and with a wiring diagram in hand, the major source of mystery is removed.
(From what you said in your post, it sounds like the coil may have some windings of the secondary shorted. I'd try replacing that first, if you can't lay hands on better electronic test gear than a voltmeter.) And always insert a resistor in circuit before you take voltage readings. Open circuit readings tend to be deceptive.
PS The ignition kill switch works by shorting the white wire to ground, either the black ground wire, or chassis ground. The white wire runs to the CDI unit and back to the magneto ignition-side winding. Shorting that to ground starves the CDI of power to pulse the coil. In effect, it shorts the magneto ignition-side coil. Simply disconnecting the white wire from the ignition switch completely should remove any possibility that's the culprit. But I doubt it as you report something happening across the coil primary. So the coil remains my #1 suspect. If there were a short from the coil secondary to ground lug, that would produce the symptoms you describe: something on coil secondary when its ground lug is disconnected/floating and nothing when the lug is tied to ground.