The most common noise that is "just that way" is the clutch rattle. Right side of engine, at idle, goes away when clutch lever pulled in.
Any other noise is not common, or at least not "just that way".
You are right to suspect the chain. Regardles of what the sprocket teeth look like, if its left side of engine, its almost got to be chain and/or sprocket. The only way to really lube an old chain is to remove it and soak it in a solvent bath and bring it back to life 1 link at a time. Better to just buy a new chain. Both age and mileage will puke a chain. Sprockets, mostly mileage.
Transmissions themselves rarely make any noise. Shifting is clunky but that's the nature of a constant mesh transmission (all motorcycles) and only occurs at the shift, not ongoing. a transmission problem usually manifests itself as jumping out of gear into a false neutral.
Buy a cheap chain and I suspect a quieter ride. If your ready buy new sprockets and a good chain and prepare for silence.
PS: The rollers on a chain can get real rattly and louder as the chain warms up. You did say you only hear it when the bike is rolling.