Hmm, I don't know about this. Let's see, Grace builds on nature so that, what you are calling "a certain natural faith", I might call "credible". Or maybe as you have it. I don't know. What I do know, is that the request is itself a result of supernatural Grace called 'actual Grace', not to be confused with 'santifying Grace'. (c.f. "No one can come to me unless the Father draw him") It is at Baptism that the habitual Grace of "santifying Grace" is received, as you state.
In any event, "Amazing Grace" is much, much, much better than many of the OCP ditleys that are sung.