FWIW, “secures” and “endures” rhyme perfectly for me (I grew up in the Midwest).
I’m not sure that the “When we’ve been there ten thousand years” verse signifies in this discussion; Newton didn’t write it.
Can’t speak for sure about the pronunciation of “come” and “home” in newton’s time, but just to point out that this is not the only hymn which rhymes the two words: “Come ye thankful people, come/Raise the song of harvest home” is another (written in 1844, says Wikipedia). So is “Watchman, tell Us of the Night,” which includes the stanza “Watchman, you may go your way/hasten to your quiet home./Traveler, we rejoice today,/for Emmanuel has come!” There are other examples too–happy to dig them up if you want them.
I would guess one of two things: either it was acceptable, if not standard, “back then” for one of the words to be pronounced to rhyme with the other (most likely “come” pronounced as “comb,” as someone mentioned upstairs); or the words didn’t ever rhyme precisely, but the similarity in sounds allowed hymn writers to rhyme the two words as a matter of convention.
It would not shock me to learn, similarly, that “snare” was once pronounced (either regularly or occasionally) to rhyme with car/bar/par/far. “Are” is usually pronounced that way, after all.
Anyway, a few other examples of words that are closer today to being “eye-rhymes” than to “ear-rhymes” that appear in hymnody: “one/throne” (Isaac Watts), “throne/none” and “home/tomb” (Samuel Crossman), “love/move” (several), “proclaim/Bethlehem” (whoever was responsible for Hark the Herald Angels Sing) “God/load”…and many more.
I agree that the verse mentioned in the OP is a little less “regular” than the others in the specific hymn “Amazing grace,” but there’s nothing unusual in a broader sense about the rhyme (or lack thereof).