Interesting, but probably beside the point. We don't have any examples of the ballad prior to the 19th century; though Child pointed to various earlier European analogues and Bronson quotes the opening verse from a 15th century carol which is vaguely similar in a conventional manner (the rest of the song is entirely different). Maybe it has early roots, and maybe it doesn't. We can't base commentary on that vague possibility.

Worth quoting, perhaps, a few lines from Ruth Harvey, 'The Unquiet Grave', in Journal of the English Folk Dance and Song Society, vol IV part 2 1941, 49-66:

"... it seems highly probable that the 'magic task' stanzas which occur in several versions are survivals of the lost troth-plight motif." [here Harvey assumes that the troth-plight redemption in 'Sweet William's Ghost' must necessarily have been present in notional 'earlier' forms of 'The Unquiet Grave'; without actually producing any evidence]. "The fulfilment of these tasks would be the condition upon which the troth would be returned. At least six versions (footnote: versions noted by Ella M Leather in Herefordshire, Charlotte Burne in Shropshire, Alfred Williams from the Upper Thames, one of Cecil Sharp's, and one of Baring-Gould's versions, and that from Gypsy Tents, by F Hindes Groom) contain this stanza, and they agree so closely as to preclude any possibility of casual interpolation from another ballad." [Harvey fails to take account of -or even to mention- the broadside editions from which all these appear to derive.] "The following is a typical form of this stanza:

'Go fetch me a nut from a dungeon keep,
And water from a stone,
And white milk from a maiden's breast
That babe bare never none.'

"The first impossibility usually concerns the fetching of a nut from a dungeon, although Sharp's version has 'Go fetch me water from the desert'. In two versions the 'nut' has been corrupted; in Groome's version to 'note,' and in the song from the Upper Thames to 'naught,' whereby the whole stanza has been turned into a negative, and has become perfectly meaningless. In Baring-Gould's version the word has been altered to 'light'."