Usually, the official reason Protestant songs are rejected is because they contain problematic theology. Of course, given the meaningless drivel that is used in its place (”Let Us Build City of God” appears to directly contradict St. Augustine, although isn’t heresy only because it’s so meaningless), that’s hard to believe. But FWIW, Amazing Grace and How Great Thou Art are in just about every Catholic hymnal I’ve ever seen, although How Great Thou Art is seldom used..
“A Mighty Fortress” was in the Missal when we got married.
Played it at my wedding. Made the priest laugh.