A number of works have noted the custom of reciting Psalm 6 at a circumcision, either after the actual berit milah or at the festive meal that follows.
This view of the superscription, claiming that it links Psalm 6 with berit milah, is hard to substantiate.
The psalm, an important text deserving its own treatment, is used as a didactic tool for understanding mezuzah, tzitzit, tefillin, and berit milah.
Because of that blanket denunciation of God's people, Elijah is made to appear at every berit milah, every newly initiated covenant, and give humble testimony of the Jewish people's loyalty to their ancestral heritage.
With this Phinehas-Elijah identification in mind, there could be another reason why the Pinhas verses are recited at a berit milah, one based on Maimonides' rationale for the law of circumcision: "As regards circumcision, I think that one of its objects is to limit sexual intercourse by weakening the organ of generation as far as possible, and thus causing man to be temperate ...