I took an Institute class a long, long time ago about the history of music in the church. If I remember correctly, Amazing Grace was included in a ~1920s edition of the LDS Hymnal, but was left out of later editions … just because, lol. Gotta have room for everything else, and songs fall in and out of favor. Plus over time it’s definitely acquired a “fundy” flavor to it (Baptist, Pentacostal, that sort of thing). Of course, so too did How Great Thou Art — one of the most loved hymns by Mormons now, I’d say — until the prophet’s wife (can’t remember which one) lobbied to have it included in the ~1950s or ~1960s sometime. Now that one sounds as Mormon to us as If You Could Hie To Kolob, so I have hope that Amazing Grace will one day make it back in as well.
Take all of this with a full shaker of salt, as my memory is slipping and I could be wrong about every bit of it. ;-)