So how did Amazing Grace achieve the kind of popularity that results in the risk of it becoming too-often heard?

There is some nice information about that in the notes to the Smithsonian Folkways CD "Jean Ritchie and Doc Watson at Folk City".

Paraphrasing...
the notes say that when Jean and Doc sang the song together in 1963 it was still little known.
Now I quote: "the version that is seminal to the current popularity of this song was that recorded Newport Folk Festival of 1963. The singers were Jean and Doc with Clint Howard, Fred Price, and Clarence Ashley. Issued on Vanguard records, their version was soon covered by..... [others]... that it was their setting of the old song that led to its popularity is easy enough to prove becuase they used the first verse as a chorus, something that had not been done before. The popular recordings that followed also used the first verse as a chorus and all show their clear and unmistakeable evidences that Doc, Jean, Clint, Fred and Tom [ie Clarence Ashley, his other name] were carefully heard by those who took up this song of faith and made it a household possession of all Americans."--Notes by Joe Wilson...

If you are tired of the tune of Amazing Grace, check out the slow air 'Mrs Jamieson's Favourite' which can be found at thesession.org