By A connection do you mean at the reg (brown wire) ? If so shouldn't that connection have battery voltage as a minimum, and say 13v plus when it's charging? Or did you measure it with it disconnected?
I'm assuming you've not had the battery out and put it back with connections reversed
I'm not sure what voltage the brown wire would have when disconnected from a battery.
Maybe someone could measure a 12v reg set up to see?
I don't think any harm comes to dynamos that are spinning but not connected to a reg. they don't generate high voltages that could damage it etc.
It does sound like the reg is duff but it's probably worth cleaning the commutator as the armature can give a reasonable voltage from residual magnetism (particularly if you are using a very sensitive voltmeter), but despite that it's still not able to give enough current to get the Dynamo going due to dirty brushes etc
Flashing the Dynamo might also be worth a try.