Now that I think about it (amazing what sleep will do for you), none of the actual hymns listed are all that old.
Amazing Grace is either late 18th century or early 19th depending on who's history you read.
A Mighty Fortress is Luther so that is 16th century.
Jesu Joy of Man's Desiring (Zzzz) is Bach, so that's Baroque.
Compare that to "Tantum Ergo" and "Pange Lingua" which are St. Thomas Aquinas and 13th century. "Veni Veni Emmanuel" or "O Come O Come Emmanuel" is 8th or 9th century. All of that is chant and all have much more than three note up and three note down intervals. And all have text which is not heretical.
Eagles' Wings' text certainly is not heretical.
It's the 91st Psalm