On the wiring diagram page 7-5 you see the Y/W wire that delivers power from the E terminal on the ignition switch directly to the coils.
So the coils always have +12 delivered to them, from the Y/W wire whenever the key is on.
That supply should be switched by the ignition switch so measure it to ground, if it is still there ignition off then the ignition switch is faultly or is wired wrong.
The black / Yellow wire goes to the igniter module and it grounded to make the coils fire.
I originally forgot that this is a Kettering ignition, not a magneto ignition.
If the coils are working properly they should spark whenever the ignition is turned on and the B/Y wire is grounded or ungrounded.
They should also be electrically isolated so neither terminal should show continuity to ground with the spark plug wire removed.
Some times there is an insulator on the mounting bolts to ensure that the coil is isolated.
As the igniter grounds the coils, to collapse the field & induce the spark in the secondary winding, if the coil is getting a ground from any other source then it becomes a poor output resistance heater.
The two small coils near the flywheel are the electronic version of a set of points.
When the magnet passes them it generates a small voltage which travels to the Igniter Module .
The Igniter Module waits a set amount of time ( milli seconds ) then closes the B/Y wire to the coil to ground which should cause the coil to fire.
The time delay varies according to the voltage the Pulsor coil generate
Being that the faster a magnet passes a coil the higher the voltage the coil generates.
So the delay in closing the B/Y wire varies to advance the spark .
In reality , the advance is simply less retard as you can slow electricity down but you can not speed it up.
As previously mentioned, a really great system, while it works and it is what makes these engines so good .
The same applies to the Kohler magnum series.
However when they don't work they are a PIA so Kohler abandoned their variable timing when they brought out the new models.
SO turn the ignition on and make & brake the Yellow / White wire at the individual coils with the B/Y terminal jumped to ground
They should fire at every make & break but the make will be smaller than the break so you might only see it with a meter.
Repeat without the jumper wire and there should be no spark
Likewise , with the ignition turned on you should get a spark every time to jump the other terminal to ground.
If the Igniter Module is working you should get make & breaks between the Black / Yellow wire & ground when the crank is being rotated.
Now because the entire triggering system is a GROUND circuit, it has to be isolated from GROUND everywhere and a short to GROUND is a very hard thing to find.
IT could be nothing more than a dirty plug on one of the connectors, a missing or squashed isolating washer.