The exciter relies on residual magnetism to pull itself up.
Needing to apply a voltage to get things started implies the residual magnetism isnt holding, you may need to flash it with a higher voltage when stationary.
Any bearing troubles can cause vibration that can destroy residual magnetism as well.
Disconnecting the AC load before running it down may also help to keep it active.
The other machine with the voltage differential sounds like it has either an open field, or shorted turns. An oscilloscope trace will identify the latter, an ohms test the open field.
Otherwise, having several identical units allows you to swap components to hit the faulty part by elimination.
Generally, voltage sag of around 20 percent from zero to full load is acceptable for these machines, they are generally set to supply 10 percent high at no load, and 10 percent low at full load..... (voltage).
These were made well before electronic AVRs were invented.
The brushes are angled away from the direction of rotation to minimise wear and reduce noise when a commutator is used.... copper segments, DC, and usually at right angles with slip rings.....AC, so maybe one of your machines was set up for counter rotation?
On some larger units, the brushes can be rotated slightly a few degrees around the commutator to alter the magnetic pole position in relation to the armature, this sets the AC voltage no load to full load change, and can be tricky to get right.
The actual position of the magnetic flux changes, leads or lags depending on load, and is the basis for the passive voltage regulation. Sometimes salient poles were used also to distort the magnetic field to further improve regulation, but probably not used on your machine.