"Singing" the Pslams is what the Catholic monastics did at everyone of their 7(?) daily services... Gregorian Chant.

The Protestant innovation was translating them to the vernacular (English in this case) and setting them into metrical forms. The English & Scottish got the idea from Jean Calvin who'd done it in French, before Cromwell's time. There have been periods of time when no texts apart from Psalm texts were allowed to be sung. Others when singing was only before or after, not during the church service.