Why don't Haza"l say Eliyahu Hu Pinehas?
My translation of the Hida in Midbar Kedemot:
Eliyahu Hanavi Zachur Latov originally was an angel. And at the time that HKB"H wanted to create the world, He said to Eliyahu and to the other angels, "Let us make man." He [Eliyahu] said before Him, "If it is good [to create man] before You, all the more so, before me [it is good]. If it is good in Your eyes, I will go down and serve before him..." He went down later on and made it so that the world would believe that Hashem is G-d. And a few days after they believed this in the world, he was brought up to the heavens, and HKB"H said to him, "You are the guardian over My children forever..." - Yalkut Reuveni on the verse "Let us make man" in the name of Sefer HaPardes.
And according to this, how sweet are the words of our rabbis z"l, who say, "Pinehas is Eliyahu," and they were careful with their wording because they should have said, "Eliyahu is Pinehas". But with this understanding, it makes sense because Eliyahu came first because he was an angel.