Of course he should

Although the lyrics and melody are not his, he makes them what they are. And the solo viola in "Black is the colour of my true love's hair" is his, not folk, although it sounds like a fiddle. He interpreted these songs and made them concert songs, it's like a photographer taking a photograph of a village: the village is not his, but the photograph, the village at that very particular small period of time represented on paper as it is, that's his. And that's what's Berio's

Berio didn't say he wrote the melody or lyrics; if he did say that, that would be wrong.
Although I don't see why we should give more credits to Stravinsky for using a well-known Lithuanian folk-tune in the beginning of his Rite of Spring,
although he supported that he never used folk tunes/themes in his music (which is proven wrong, with pieces like this, Pribaoutki and the 3 pieces for clarinet)
