Whence the 'OK' sign?

I know the generally agreed-upon etymology for ‘O.K.’ is ‘oll korrect.’

I also know one of the fake etymologies is that it meant ‘0 K.’ short for ‘zero killed.’ The way that one was explained to me, a soldier on the front lines would give a status to a superior with that silent hand sign - holding up an ‘0’ to indicate zero killed.

Is it possible that OK was already in use, but that that’s where the hand signal for ‘OK’ came from? Or is there another explanation?

I found a reference here to a book, Gestures: Their Origins and Distribution, by Desmond Morris et alii (1979), which purports to describe the origins of:

There are many advertisements from the 1930’s showing chefs and others giving the OK sign to indicate that the food/meal/whatever was OK/great/perfect.