On the smaller alternators Briggs usually fits the diode right at the output connector. You can see it in your picture: red/white and black wires into the white/clear connector, with a diode in the red/white wire. The red/white wire with the diode is connected to a yellowish-white wire on the tractor-harness side: this is the DC charging current to the battery, and the tractor electricals in general. The black wire from the engine's alternator into the connector is not being used at present - it carries AC, and would normally be connected to the headlight switch on the tractor harness side of the connector. This coincides with Gadge's Figure 11, but the feed wire to the diode is red and white instead of red.
Greenfield, like many other tractor manufacturers, does not use the AC supply circuit. My guess as to why they do not use it is that they want the headlights to be available for use when the engine is not running. Of course, the AC supply will not produce any output when the engine is stopped, which to me seems like a good thing, since it prevents the battery being discharged foolishly by inappropriate use of the headlights.
When you replaced the perished fuel hoses, did you remove the carburetor float bowl and clean the inside of it, plus clean out the anti-afterfire solenoid, and the main jet? The material from the perished hoses has to have gone somewhere, and those places seem likely destinations. Note also that the anti-afterfire solenoid on Briggs engines has been a very unreliable component, though at the moment I can't relate a problem with it (other than its fuel passage being blocked with junk) to your symptoms. For diagnostic purposes people sometimes remove it temporarily, screwing in a suitable very short bolt instead, to see if that fixes something. If it does, the solenoid has to be replaced with a new one.
