I wouldn't presume to comment authoritatively on the particular scriptures, not while I have this headache. But I checked the Rom 15:16 (with which I was not familiar) and "priestly service" sure seems right to me, with the "iero" stem as part of the word.
I do very much like your general theme, which I will mangle thus: We are all priests and the church has an order of priests which most definitely is NOT a continuation or a ver. 2.1b of the OT priesthood.
And we are, and the Church does have, priests because of the mystical (and, I would keep insisting, proleptically apprehended eschatological) union with Christ in the Spirit, the union which makes the Church His body.
In and because of that objective Spiritual union, by God's grace, which is always more than we could imagine or pray for, we may make feeble, half-hearted, and self-conscious offerings of our "works and prayers, joys and sufferings" daily and take confidence that these little things of ours are swept along in the great torrent which is the Passion of Christ.
And as Christ and His Passion, Death, and Resurrection (and all the rest) somehow stand as the forming standard for our offerings, the Eucharist is so united to Him and His work that it is rightly called the source and summit of our faith.
And so the sacred presbyterate, with the college of Bishops and that of deacons, have a creating (in an intermediate way) and a "norming" relationship with the priesthood of all believers. We all offer Christ and we all are that which we offer. The whole Church, the laity in a special way, make that offering mystically. In addition, the presbyterate makes it sacramentally.
How's that?
Not an answer just church teaching