The earliest extant English ballad that contains the rose/briar motif is indeed Fair Margaret's Misfortune etc. a variant of Child 74 printed by Sarah Bates at the Sun and Bible. I have seen estimates of the date of this as 1700 and 1720 but Wm Chappell who is usually very reliable with these dates gives it as 1685. The Bates family were printing over quite an extensive period but the S. for Sarah and the address make the dating more precise.

Here are the final 3 stanzas as printed. I've given them letters A B C for later reference.

A
Margaret was buried in the lower Channel (Chancel)
Sweet William in the higher
Out of her breast there sprung a rose
And out of his a briar.

B
They grew as high as the church-top
Till they could grow no higher
And then they grew in a true lover's knot
Which made all people admire.

C
There came the clerk of the parish
As you this truth shall hear
And by misfortune cut them down
Or they had now been there.

Looking at ALL of the English language uses of the motif I see nothing that couldn't have derived easily from this, including the Scottish birk variants.

Jamieson's lengthy notes (1814 'Illustrations' on the net in Google Books) on Ribold and Guldborg, including much information on other Nordic variants such as the HildeBRAND versions, were available to whoever put together EB. Why would Scott or your proposed writer of DT take only part of the story from EB?

Whilst Child does indeed mention the connections of 74 with 73 'Lord Thomas and Fair Annet' I don't think he mentions that on the Douce copy someone has written 'To ye tune of Lord Thomas and Fair Annet'.
The early broadside copies of 73 don't contain the birk/briar motif but the later Scottish copies do, the earliest being one sent to Percy in 1765.

Of the other ballads that have picked up the motif, Child 76 'The Lass of Roch Royal' has a version that can be traced back to c1730, and one version of Child 75 'Lord Lovel' can be traced back to 1770. All of the others can be traced back no further than the 1820s and are likely derivative, not necessarily directly from FM&SW, but ultimately (IMO).

When we see the Scottish 'birk/briar' in stanza A, stanza B always has the plants growing near and 2 lovers dear, lacking the 'true lover's knot', whereas the earlier (IMO) 'rose/briar' is always consistently followed by the 'true lover's knot' in stanza B.

The latter combination is remarkably consistent in Barbara Allen versions considering the large number of variants, even including the 5 English versions given in Bronson that have the motif. The who gets the rose and who gets the briar situation is fairly consistent with the behaviour of the baddy/goody of the piece. In BA the rose is frequently red but there are 3 versions where it is white. It would appear that Campbell's politics have spread into America!

Stanza C is rare in oral tradition but crops up in the A version of Lord Lovel where an old woman cuts the plants down; Lady Alice A version has a priest do the honours (B has the wind doing it). A version of Fair Janet sent to Scott by Laidlaw has a French lord pull up the plants (possibly what inspired Scott to make Black Douglas chuck them in a pond). Incidentally, none of these seem to be implying that the cutting down/pulling up was a vicious act, except for Scott.

I'll hang onto my notes in case anyone wants further info, but it all looks fairly straightforward to me, nothing astounding.

For a much more in-depth analysis of this motif and many others I can recommend Fleming Andersen's 'Commonplace and Creativity'.