Yes, the black wire is the kill wire. You need to be sure it is not accidentally grounded anywhere, including at the switch. If that is the case, there seems to be a fault in the module. The term magneto includes the magnet in the flywheel.
Most of the mowers around have either Briggs & Stratton or Victa engines, so parts for them practically grow on trees, or at least on nearby nature strips. Your Suzuki is likely to be in a different situation, though. A replacement may be relatively expensive (compared with nature strip harvesting, at least). It might be worthwhile seeing if your local mower dealer can test it for a nominal fee.
It contains two transistors, a normal ignition-type coil, and not much else. However it seems to be encapsulated, so I don't think there is much that can be done about repairing it. Do be sure the ground wire (yellow) is making contact, though. My preferred approach in this situation if I can't get a free test on the suspect component is to try to borrow a replacement and see if it does the trick.
