I see the distinction, and #14 is an excellent illustration, but in this particular example I find it hard to see much difference between
X going gentle (where 'gentle' is some sort of state of X)
X going gently (where 'gently' is a characteristic of the going).
If X is gentle, the going will be gentle, and if the going is gentle, X will be gentle during it.
The following line of the poem:
Old age should burn and rave at close of day
does sound to me like it's talking about the going.
So, without wishing to dismiss the gentle/gently distinction as discussed above, I nonetheless read/feel/hear 'gentle' as describing the death, more than the person dying.
The choice of 'gentle' would then have to do at least in part with its sound and feel as language. 'Do not go gentle' sounds a little stranger, rougher, than 'Do not go gently,' which would be the more 'ordinary' way of putting it.