If you’re an urban, college-educated nontheist singing 19th-century Southern rural religious music, there are two options. You can take out the weird and uncomfortable bits and make it be about friendship:
What wondrous love is this, O my soul, O my soul?
What wondrous love is this, that brings my heart such bliss,
and takes away the pain of my soul.When I was sinking down, sinking down, sinking down
When I was sinking down, beneath my sorrows ground,
friends to me gathered round, O my soul.(Unitarian Universalist hymnal)
Or you can sing it un-retouched:
What wondrous love is this, O my soul, O my soul,
What wondrous love is this that caused the Lord of bliss
To bear the dreadful curse for my soul!When I was sinking down, sinking down, sinking down,
When I was sinking down beneath God’s righteous frown,
Christ laid aside His crown for my soul.(anon, published 1811 in Lynchburg Virginia, A General Selection of the Newest and Most Admired Hymns and Spiritual Songs Now in Use)
The obvious thing for a blog post to do here is to make a point about one of these being better than the other, but I’m not sure. I find it kind of icky to tidy up and happify music that is at its heart deeply concerned with Hell and who is going there.
But the hipster approach, to sing the original lyrics with no connection to the value system of people who made them, also feels weird. If this weren’t white Americans singing music by other white Americans, it would be called cultural appropriation.
But I don’t think anything bad is happening, or at least not worse than leaving urban atheists without this weirdly beautiful music.
The whole sky 
Oh, this is such a fascinating and complex topic. The interesting thing about many Sacred Harp groups outside of traditional singing areas, is that even if many of the singers don’t share the faith of the traditional singers and the hymn writers, they are in fact paying a lot of attention to where the music comes from and striving to engage in traditional musical and community practices in many respects. (Interestingly, some of the groups with the weakest connections to traditional practice are not “urban hipster” singings, but rather Folk Revival-era groups founded by Boomers decades ago.)
And even within the beliefs and practices of “traditional” or heritage Sacred Harp singers, there’s quite a bit of theological diversity. Throughout its history, singings have attracted people from various Christian denominations (which was about as much theological diversity as one could expect in, say, 1880s rural Georgia), and there is a strong cultural norm to “leave religion and politics at the door” when you come to a singing.
There’s also not a bright line between “urban [atheist] hipsters” and “traditional [Christian] singers” — “urban hipster” singings often have a significant number of Christian singers, though some may come from denominations or belief systems that are quite different from those of traditional singers (Catholics, for instance, or various progressive Christian denominations). And many non-Christian singers still feel a deep and meaningful connection to the hymn texts, even if it’s not rooted in Christian belief specifically.
I offer this (perhaps excessively detailed) context not to undermine your point, but to call out that many non-traditional Sacred Harp singers do, in my opinion, quite a good job of understanding and respecting the tradition they’re drawing from.
Thanks for catching a lot of the nuance I missed/oversimplified!