"And, I repeat, why not say that the besom-maker sang (or recited) it to Dixon?"

May I suggest a possible reason? Let's go back to the definition we have been using, though it isn't ideal:

'By mediation I mean not just simply the fact that people passed on songs … but that in the very process of so doing their own assumptions, attitudes, likes and dislikes may well have significantly determined what they looked for, accepted and rejected.'

Harker has been discussing the various sources that Dixon made use of. He speaks approvingly of the few occasions when Dixon used singers as a source and the besom-maker was one of these. It may be that Harker wanted at this point to emphasise that the humble besom-maker, just like the other kinds of source, will have had his own attitudes and assumptions about the song. These will be part of the culture that Harker argues was too often ignored or 'manufactured' by collectors.

I can see one might imagine that Harker thinks of a performance of the play as in some sense 'authentic' and that the man will be 'mediating' that 'authentic' version, but I don't think we need to think that way in order to come up with a reason why he uses the term at this point.

I'm not sure that Harker is saying that there is any 'unmediated' version, if this makes sense? Because each and every version will be an interpretation?

Til

But I do agree that it can be a bit irritating when he repeats words.