Well, it is obvious even to me, a non-concertina player, that Peter didn't do complex tune playing that required moving the fingers rapidly among the buttons. (Even that 'solo' at the end of " The Fox Jumps Over the Parson's Gate" is more in & out on the bellows than playing a tune)...but, that said, it IS effective, musically showy, and fits the way he did the song.
   It is not necessary to be an Alistair Anderson to produce a melodic and relevant accompaniment! (I know someone who plays guitar largely with thumb & one finger, yet it sounds fine as a melodic backing to songs.)

   I can't see why it's even relevant to suggest that a 'style' is somehow a 'flaw'. Peter was not trying to play jigs & reels, but merely to help the song along...and he did so pretty well.