If the old priesthood is abolished, what is the point of having Catholic priests? Before Christ, there would be two tabernacles in the temple: an outer and an inner tabernacle. Priests would minister at the outer tabernacle by caring for the lamps, by burning incense to God twice daily, and by giving weekly offerings of bread. Once a year, the high priest of those priests would sacrifice an animal, and then he would dare to enter through the second veil into the inner tabernacle. There, he would sprinkle the mercy seat, the place of God's presence, with the sacrificial blood, and would then invoke the name of the Saviour God in atonement for the sins of Israel. Then he would leave.

When Christ offered up His sacrifice, He did not go to the man-made outer or  inner tabernacle. He passed through into heaven itself to the seat of God the Father. There at the throne of God He sat down at the right hand of the Father as our most perfect intercessor. Our High Priest went up to that most sacred place to ask God to forgive us, and He remained there. That is how Jesus is even now serving as High Priest.

Since we no longer need high priests to intercede for us by way of repeated bloody sacrifice, why do Catholics have priests? They are obviously not re-sacrificing Christ when they say the Mass, so what do they do as priests, and why are they called priests if they are not (to put it bluntly) killing something?

(Answers from other denominations (Orthodox?) that have compatible reasons for having priests are welcome.)
They are obviously not re-sacrificing Christ when they say the Mass


It's true enough that they are not re-sacrificing Christ. But in a mystical way the Catholic priest acts in persona Christi to offer, in an unbloody way, the very same sacrifice that was offered in a bloody way on Calvary. Here is the modern Catholic Catechism (paragraph 1367), quoting the 16th-century Council of Trent (my emphases):


  The sacrifice of Christ and the sacrifice of the Eucharist are one single sacrifice: "The victim is one and the same: the same [Christ] now offers through the ministry of priests, Who then offered Himself on the cross; only the manner of offering is different."


Thus a principal point of having priests, in the Catholic Church, is to continually make the unique, unrepeatable sacrifice of Calvary mystically present to us here and now. Our priests, like Christ himself, are of the order of Melchizedek, not the order of Aaron. Indeed there is in a sense only the priesthood of Christ -- as St. Thomas Aquinas says (and the modern Catechism quotes, paragraph 1545):


  Only Christ is the true priest, the others being only his ministers