If it were me, I'd hook the black/white wire in the original harness to the #2 connector on the alternator rather than jumping it to the alternator's output as shown
Here's where I think your problem is coming from - the black/white wire feeds the alternator's internal regulator from the ignition switch. The regulator then energizes the alternator's field without which it will not work.
No output means no voltage to energize the field, which means no output, which means no voltage to .... you get the general idea.
And the question in your mind about now is how come his own works and yours doesn't - it's called residual excitation - sometimes there is enough magnetism left in the winding cores to get it started and sometimes there isn't.
Well, theoretically if the wire from the positive terminal to the battery is connected to the post on the alt and the #2 wire is also connected to that post then the field coil should be energized directly from the battery, mainly just bypassing the ignition switch in that case.
BUT, if my original issue happened to be with that 60A fuse or one of the charging wires, then power would never flow from the battery to the alternator to energize the field coil Or to charge the battery once the motor is running. That would definitely explain why the old alt tested good on the machine, didn't turn on the charge warning light, but also didn't charge the battery. The old alt was getting field coil power from the main power bus through the 2-wire connector and was putting out enough power to keep the charge warning light off, but power was not getting back to the battery.
I think my next project might be to bypass or replace that 60 Amp fuse, possibly re-run the wire from the battery to the alternator.