I thought I might add the full arrangement of Martyrdom from The Bristol Tune Book. Sacred music isn't my line, so I don't know how standard such arrangements are; but to my ear at least, it sounds, if anything, more akin to Amazing Grace as we know it today than does the pure melody line. A little improvisation and you have it. Of course, the familiarity of the metre may to an extent deceive the ear, perhaps.

See what you think. Multi-line abcs can be a bit messy, so this time it's a midi file:  Martyrdom: Bristol Tune Book arrangement, 1863