I personally think a 16ga wire will operate a 15a O/C device if it was made up properly and even though it doesn't meet current code I would certainly use it but I agree it doesn't meet the current code for an EGC. The legal fix is going to be a GFCI. Perhaps these days a GFCI/AFCI device is the way to go if you can find one to fit the existing panel. Since that RX screams "Happy days" (1950-65) I doubt the panel will take any new style breakers at all unless it was replaced along the way. You may be tracking down the first receptacle on the circuit and using a device type GFCI.
My house has the 16ga EGC in the original wiring but I figured out the boxes are made up almost like you would see today with the grounds twisted up tight, connected by a screw to the box and a pigtail to the receptacle so I have been treating them normally. The kitchen and both bathrooms were renovated and the wiring upgraded. There are really not that many 3 prong plugs used in the rest of the house anyway when you think about it. The garage was built in 1970 with full size ground cable.
