Reading the above I have to admit to loosing the plot a bit. The OP would seem to know what he’s talking about but it’s as well to understand a few basics.You don’t actually mention the brands of engine , so I’m assuming Briggs.
The ignition is entirely self contained and does not require any external power ( the OP already touches on this and its expensive consequences) The only connection between the ignition coils and the ignition switch is one terminal that grounds it when the switch is off. Buried in one of the Briggs manuals is an instruction to perform 50+ on and off checks that said kill terminal never becomes live due to an internal fault.
In addition to the kill wire from the coils to the kill terminal of the switch , there may also be a diode harness connecting the two coils which stops them interfering with one another . Failure or absence of this harness can produce some strange consequence, such as running on one cylinder or continue to run on one or with the key in the off position. Again, buried in one of the manuals is a little chart giving the various combinations of symptoms from diode failures .
For what it’s worth at this stage, my best advice is – disconnect the kill wire from the coils and add your own kill wire, which by grounding, will be your only way of stopping the engine . Check out the diode harness and renew if faulty . Then fire up the engine and check that it runs on both cylinders and stops on both when your kill wire is grounded. If it performs correctly you then know that the ignition is ok.
Now don’t reconnect the original kill wire until you have ascertained that none of the alterations, additional relays etc, are introducing a spurious power feed to the ignition kill circuit.
Then with the aid of a wiring diagram, Check out the charging circuit. Note that the square box with the yellow wires is the rectifier regulator and needs 9 volts to operate . If there’s a flat battery and the machine is jump started , the machine will never charge. Start with a fully charged battery. Also note that the rectifier regulators are specific to the alternator type and capacity. Again, there are charts showing the various options of alternators and their appropriate rectifier regulators.
