Well, there's this thing called 'conversation' where people talk to each other at home. So no hymns required.
I wonder why you are so obsessed with hymns, which are inherently Christian, and why you seem to be completely unaware of the existence of folk songs, which would absolutely serve the same purpose? I agree with the benefits of communal singing, by the way - the research on it is pretty clear, and I was working in mental health research at the time the first studies into it kicked off.
But my DS, who has depression, is in a choir. It has benefited him. And they sing a couple of religious items as part of their repertoire, but they also sing ABBA classics (arguably a part of British cultural capital), folk tunes and old school protest songs.
So instead of obsessing about hymns, we could go with 'Blowing in the Wind', 'The Skye Boat Song', 'Scarborough Fair', 'Greensleeves', 'Drunken Sailor' and any one of hundreds more.
When it comes to the Remembrance season, schools intending to sing at memorial services could do this thing called 'learning a song, and rehearsing it, then performing it'.
Your idea that religion and religious songs could in any way be a uniting force is fully ridiculous. Religion is being used to divide right now and we should not fuel that fire.