But at the moment... i can't see a need for it.

Lots of small applicances and such have only two prongs yet carry certifications like UL. I have no idea what the criteria is for requiring a 3-prong grounded plug in consumer goods. In general terms, having a 3rd-prong ground wire that really goes all the way back to ground/earth provides a harmless path for bad-bad-bad electricity to follow in the case of a short rather than having that bad electricity follow a path to ground through your forearm and thorax. That's certainly a good thing.

Those little 3-to-2 prong kludge adapters have that little tab on them that ties the 3rd prong into the body of the electrical outlet through that little screw. But, as you suggest, that only helps if the outlet body is actually grounded. If it's an older, ungrounded 2-wire setup, then what's the point?

I think you can do no harm by running the green ground wire to your CH-2 case. Some day, you may plug it in to a real 3-prong outlet and may save yourself the embarassment of getting zapped like that bad guy in Goldfinger.

When I get confused about black and white -- as happens with increasing frequency -- I just make sure to use one of these .