it is unlikely, to say the least, that the spread of RUS at singarounds in the US has any link to church attendance.

I think you misunderstood the original point. What we've been suggesting is that the difference between the US and the UK in this respect may have something to do with the difference between the two countries in levels of church attendance. As Marje said,

although communal hymn-singing is not a common feature of modern life in the UK, many of the people over 50 who insist on reading every word from a book or file have, like me, spent their formative years singing from hymn books like this, and don't know how to do it any other way.

Perhaps a country with a higher proportion of regular church attenders (past as well as present) will have a higher proportion of people who "spent their formative years singing from hymn books", and consequently a higher proportion of people who find it natural to sing from a book. That's all.