It was the 70s, and in North America at least grounded outlets were a fairly new thing, only standard for about 15 years. Not much was made with an earth, or double insulated. Can modify that if you want and are knowledgeable but it can introduce different weird problems.Yes, often it is. However, I don't consider that to be good practice unless it is double insulated and not connected to mains earth.
On this one i think the fix is to evict the modern parts and put them in an external box. Shields and grounds are likely to block the RF and make it not work.

