No, it does NOT teach Christ "as our priest" with regards to the body of Christ.
It teaches Christ "as their priest" with regard to the nation of Israel and the promises made specifically to them.
Perhaps you'd like to attempt an answer as to why Paul, the apostle of the gentiles, never ONCE uses the words "priest" nor "priesthood" in ANY of his epistles. If "Christ is our priest" (and by OUR, I'm referring specifically to the body of Christ), then why does Paul never mention that in ANY of his THIRTEEN epistles. Seems like of glaring omission.
Your logic is completely faulty. You do not understand the dispensations of God. You do not understand the promises that God made to Israel. You do not understand Christ according to the revelation of the mystery.
It looks like we're approaching this question with two completely different view points. My view is that only believers are citizens of Israel, and that the promises God made were made to those same believers. Non-believers have been broken off and are no longer worthy of the promises, regardless of genealogy.
Perhaps we should discuss that in another thread.