Last night my (acapella) chorus director commented that one of the reasons we don't use a piano even for rehearsal is that it has an 'even-tempered' scale and when we fine tune a chord using just the human voice we use a slightly different scale which doesn't quite 'match' the piano. Maybe one of our musicologists could elaborate.   I heard a young quartet sing recently where one child had a bad voice day and their teacher 'helped' by playing a keyboard. The teacher RUINED an otherwise great song, didn't seem to be able to hear that they were tuning without her.

You are correct about 'freedom', I think. You feel freer singing. Without instruments the vocalists can control all the elements: rhythm can be more irregular, improvising leads can wander the melody around at will, and as above, the harmonizers can sound chords their own way.

My own pet theory however, is the folk (as today) liked to form new splinter churches all the time and if they were poor there was no piano at first and by the time they could afford one their music was so together they don't need one.