Pope Pius XII misfires on citing the author of Hebrews as Saint Paul. That anonymous author is indeed the source of much of New Testament-based teaching on the high priesthood of Jesus Christ. As is Saint Augustine, as cited in Mediator Dei 76:
76. Now the Apostle of the Gentiles proclaims the copious plenitude and the perfection of the sacrifice of the cross, when he says that Christ by one oblation has perfected for ever them that are sanctified.[Cf. Heb. 10:14] For the merits of this sacrifice, since they are altogether boundless and immeasurable, know no limits; for they are meant for all (people) of every time and place. This follows from the fact that in this sacrifice the God-Man is the priest and victim; that His immolation was entirely perfect, as was His obedience to the will of His eternal Father; and also that He suffered death as the Head of the human race: “See how we were bought: Christ hangs upon the cross, see at what a price He makes His purchase . . . He sheds His blood, He buys with His blood, He buys with the blood of the Spotless Lamb, He buys with the blood of God’s only Son. He who buys is Christ; the price is His blood; the possession bought is the world.”[Saint Augustine, Enarr. in Ps. 147, n. 16]
The encounter with Christ, priest and victim, must be “vital,” as Pope Pius expresses it. How can an act of death be life-giving? That is a paradox Christianity has always been comfortable with. We don’t always acknowledge other paradoxes, do we?
77. This purchase, however, does not immediately have its full effect; since Christ, after redeeming the world at the lavish cost of His own blood, still must come into complete possession of the souls of (people). Wherefore, that the redemption and salvation of each person and of future generations unto the end of time may be effectively accomplished, and be acceptable to God, it is necessary that (people) should individually come into vital contact with the sacrifice of the cross, so that the merits, which flow from it, should be imparted to them. In a certain sense it can be said that on Calvary Christ built a font of purification and salvation which He filled with the blood He shed; but if (people) do not bathe in it and there wash away the stains of their iniquities, they can never be purified and saved.
The link from Good Friday to baptism has long been a part of Christian tradition. Some ancient fonts are six-sided, suggesting the sixth day of the week. The Romans 6:3ff citation used at the Easter Vigil in the modern Roman Rite asks that question: are we aware that baptism is an entry into the death of Christ?