Everything about it can and has been argued against, from the idea that there is an overarching "English Folk Music" on--

As to part playing, if there was evidence of part playing, there was part playing--you can't rule it out just because it wasn't written out--even classical music of certain periods was written with the understanding that chordal accompaniments would be improvised, and often by a group of musicians. And folk musicians, now as always, work without written parts-