It doesn't happen often, and it will put me at odds with some fellow cave dwellers, but I can't quite agree with Michael Voris on this one:
A couple observations...
First, set aside the dogmatic issues for a moment, it seems he suggests tossing away the hymn because of its Protestant roots. This strikes me as contradictory to the Pauline command to test everything and retain what is good.
Second, despite its Protestant roots, as one of the commenters points out, it is not hard to give a Catholic reading of the lyrics that is dogmatically correct. I suspect this is how the song was approved for liturgical use. While understanding an acceptable sinner's-eye view of himself as a "wretch" because of his sin is straightforward, I admit, that an alternative interpretation of the "hour I first believed" phrase strikes me as a stretch, and I usually don't sing it accordingly.
Regardless, so long as the song remains approved, it will get used, which means it will provide ongoing opportunities for a "catechetical moment." What would be one easy way to prevent, or correct, any misperceptions regarding the meanings of the lyrics? Have priests and deacons include the distinctions in a homily about Grace. How do I know? That is how I learned it. Not to mention, one effective method of cultural change is to adopt and convert the meaning of something held dear (like a "national anthem," for example?).
And lets not forget, people like the tune, including me. A nice melody, an acceptable Catholic interpretation, opportunities to teach and convert; sure seems like plenty of good to retain to me.
As for my part, I will have Amazing Grace at my funeral, omitting the "hour I first believed" verse.