Well its pretty simple:
The exciter is simple a DC generator built on top of the Ac windings in the armature. Its purpose is to supply a lot of DC to the field coils, and by a lot I mean full saturation. This type of generator does not need a regulator because the exciter makes so much flux its assumed it has enough to provide decent regulation ( about 10% maybe 15% full load to no load not bad by standards of the day )
So if you have a look at your brushes you will see there are two sets. The slip ring set draw the AC power the DC set are for obviously 12 volt DC ( more actually about 18 to 20 before the regulator for he charging system)
When you apply DC supplied from the battery to the exciter it will motorize and spin the whole generator. This is exciter cranking…
Onan wiring is a little quirky in that you tend to ground things to turn them on and off.
A photo of your controller inside will help and I can identify the parts…
There will be a starter solenoid, a couple of relays for control and a couple of relays for the battery charger. There will be a few resistors for controlling charging current.
No cranking or slow cranking has one of two causes if the controls are working one is the brushes are worn, sticking or broken and the commutator is in poor condition, worn dirty ect.
If you can get some photos of that I can probably make some better guesses.
The most common problem is a weak battery. These things really draw a lot of amps to crank and they are on par with a lot of cars with how much they demand of a battery
So next from you I need the model and spec number so I can find some prints. and photos of the condition of the wiring in the control box and brushes and slip rings.
We will work at this…
It has a the makings of a useful thread, thank you for asking these questions.
Armature in real good shape ( this one is three phase could 4 slip rings 3 line 1 common Add info it could be a re connectable type that is 120 - 240 or 120 only I can’t tell from the photo)
Here’s some field coils without the pole shoes or frame
Here’s a manual I just found, we need more details now on the model and spec to find the correct one.