In general, (and this is a gross generalization) magneto systems have two coils under the flywheel: an ignition primary coil and a charging coil. (Sometimes both of these are encapsulated into a single piece.) There is also a secondary ignition coil (or HT coil) outside the flywheel, that looks like a small version of the ignition coil on a car.
The current from the charging coil goes through a rectifier and into the battery. On some outboards, there is no voltage regulation, on others there is. In yet other outboards, the system is more refined, with an alternator that works more like that in a car. There are all sorts of possibilities, so you really need a schematic to see what's going on in your engine. But in any case, I am not aware of any system which uses a single coil in the magneto to supply both ignition and charging.
Almost all electric start outboards have separate starter motors, very much like the starter motor on a car.
As a pure guess, I'd suspect wiring first, the rectifier second, and the charging coil third. The charging coil can be tested with an ohmmeter, as can the rectifier. The charging coil should have a fairly low resistance (maybe 3-10 ohms) and the rectifier(s) should conduct in one direction but not the other. Wiring can be checked visually and with an ohmmeter.
Ignition is separate and unrelated.
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