what is the voltage with no engine running?
max voltage would be about 6.9, after that the battery starts to boil off the water.
did you do the test with engine revved up or idling?
I would look for an increase in voltage from non running to running. give it a few minutes to make up for the juice it used to turn the starter when starting, then run the test. you could try turning on the accessories like lights and heater and see if it keeps up.
if you disconnect the battery from the system then you may need to polarize the generator.
here is a delco generator on a tractor where he shows how he did the reg wiring and polarizing.
https://www.youtube.com/watch?v=7CRtUlgIgyU
here is a vid showing the delco gen on a tractor where she shows how to test
https://www.youtube.com/watch?v=II-6nOw28J8
a generator is a generator. there is a field connection and an armature connection.
you may wanna check the brushes for length and tension against the armature and also the armature to see how the contact surface looks. also the contact points in the regulator to see how they look. poor connections at either will result in inconsistent or non existent charging. a generator shop could put the thing on a bench and test it but should also have the regulator so they can set the thing up for proper output. they would also put the armature on the "growler" to test for opens or grounds in the wiring on the armature but that requires disassembly so gonna cost you cashola. at that point it turns into a rebuild usually. maybe an upgrade to a 12v alternator if that is the case as you would get reliable output. also check and ensure there is a good positive and negative connections from the battery to the generator and volt regulator. that means good body grounds and a good engine ground as well as battery terminals and cabling. I have seen old cabling cause voltage drops which will affect the whole system.
here is a vid showing some generator explanations and testing you can do yourself. english car site but a generator is a generator
https://www.youtube.com/watch?v=RjLW_eb3D-c