If it has a diode in the wire, I expect that will be a low current alternator. The mower's wiring diagram shows two black wires out of the alternator, going to a plastic connector block. It shows one red wire coming out of the connector on the mower side. The other wire from the engine would carry unrectified alternator output (i.e., A.C current) for the headlight circuit if there is one. I expect that the diode will be in one of the engine-out wires, on the engine side of the connector:

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If the wiring diagram were correct, the diode would normally be in the black wire that is connected to the red wire, not the unused (AC) black wire.

Unfortunately, you say that there is only one wire coming out of the alternator, and it is red. This would mean your engine is not the one shown on the diagram. It sounds as if you have the Figure 6 alternator:

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If you have this alternator, the diode is in the back of the engine side of the connector plug that attaches to the matching socket on the mower side of the connector. If that diode is defective, you must replace it with one of at least 3 amps rating, preferably a bit more. If you get it from an electrical shop it will probably cost you between $1 and $2. If you buy one as a Briggs spare it will probably cost a bit more, but will be genuine. I agree you can test it by checking for AC voltage upstream of the diode, and DC voltage downstream of it, when it is disconnected from the mower wiring. Obviously when it is connected to the mower, there will be DC voltage on it from the battery, whether the engine is running or not.

At the moment I don't know why you are so convinced the diode is faulty.