The reason that I have heard from Catholics who think that Amazing Grace is inappropriate for Mass is that they believe the hymn implies “once saved always saved (OSAS),” which is contrary to the teachings of the Catholic Church (and also contrary to quite a few Protestant denominations, including some evangelical denominations).
Also, they believe that the hymn implies that a person is “saved” in a one-time act of “belief” ("how precious did that grace appear the hour I first believed), rather than over a lifetime of trusting in Christ and remaining loyal to Him.
I think that the hymn’s words are general enough that we can infer those things, or we as Catholics can infer something quite different.
For example, Catholics believe in sanctifying grace, and we can choose to sing the hymn and keep this sanctifying grace in our mind.
Also, many Catholics will testify of a point in their lives when their faith all made sense to them, when their Catholic faith became real and personal, and when they made a conscious, mature decision to follow Jesus Christ as their Lord and Savior, and to follow the entire teachings of the Catholic Church to their best ability. That would fit with the lyrics 'how precious did that grace appear the hour I first believed."
I love the hymn. We sing it fairly regularly in our parish, but we also sing many traditional Catholic hymns, so there is a nice balance and plenty of musical catechesis. I don’t think too many people in our parish are in danger of being led into OSAS through the occasional singing of Amazing Grace.