It has been accepted fairly broadly, although not by everyone, and gives melodies for the entire body of verses. A very small selection is performed here. The age of the notation is one topic under dispute. The oldest manuscripts actually date only to c.900, although the music itself is said to date to the era of King David. It is conjectured that it was recopied from older sources, and there is some discussion of the notation in written sources which might indicate this possibility.
Biblical scholarship is, of course, a highly charged and controversial field.
The accompaniment is more modern in conception, based on some descriptions of instrumental ensembles in the reign of King David. I find it rather dubious, but not without artistic merit. The diction in the melodies also strikes me as distinctly modern, although the idea of metrical accent is contained in the notation, as far as that goes. While the melodic movement itself may very well accurately reflect the original psalms, the overall production certainly contains many other elements especially in the accompaniment and resulting harmonies. These have appeared on Harmonia Mundi and Alienor labels.