A commenter asked below, "...I wanted to source your opinion if I may - when the Lord walked the earth, what was the role of the High Priest in the Temple? Was the Tamid still pleasing to the Divinity? I'd appreciate some thoughts on this - or could you refer me to some commentary on it?"
Now, I must admit not to be too well-informed in this area of theology, but it seems to me that Our Lord certainly recognized the High Priest, and so did St Paul - in both cases rejecting them for their hypocrisy, but all the while acknowledging their office.
What of the ministry of the High Priest? Well, he and all his line back to Aaron were foreshadowings and types of Christ, the One Supreme and Eternal High Priest of God's religion: insofar as their deeds fulfilled this prophetic role, and their Legal duties were rightly carried out, their ministry was as a pleasing odour ascending unto the Most High, even down to the days when Our Saviour walked the earth.
According to the Jewish Encyclopedia, the term tamid refers to the manifold ritual duties performed in the Temple. Until Our Lord by His Sacrifice brought the times of the Old Testament to their end by establishing in His Blood the New and Eternal Covenant, in accordance with the prophecy of Jeremias (xxxi, 31ff), it seems to me that the ceremonies established by the Lord in the Law given to Moses were still in effect (just as Christ testified to its enduring status by telling the cured lepers to shew themselves to the priest, in accordance with the Law), and so had to be carried out that God be worshipped, praised and glorified aright.
The Epistle to the Hebrews states in chapter nine that all the accoutrements of Mosaic worship were representations of the heavenly tabernacle: it was only when the true tabernacle, the altar on high, was entered once and for all by Christ offering up Himself, that the Temple veil was rent, signifying that the earthly prefigurations of the Divine Liturgy had reached their term. The worship of Israel of old looked back to the delivery of Israel from Egypt but moreso forward in mystery to the Cross, of whose total Redemption the Exodus was a foretaste; the worship new-established by Christ at once looked to the Cross, and forward to the End of Time, when (as the prophets foretold) the elect shall join in the Messianic Feast. "Ancient types have long departed / Newer rites of grace prevail."
From A.D. 33 to 70, there remained four decades during which the former rites were nonetheless still celebrated, despite many in Israel professing their acceptance of Our Lord as the Messiah, many even of the priests coming to believe; the Acts of the Apostles tells us that for the moment they still attended upon the Temple worship, since there was as it were a time of changeover intended for men to grow accustomed to the new order, and it would have appeared scandalous to the people if the Temple rites had been abandoned forthwith without explanation: ideally, cut to the heart by the sermons of St Peter and the rest, not merely three thousand but all Israel would have been baptized, and the worship of Israel transformed according to Our Lord's replacement of the Legal requirements with His Eucharist. (It goes without saying that the pious chant of the psalms may well then have segued into what was later regularized as the Divine Office.)
But, mystery of iniquity! many in Israel - even after the Resurrection of Christ, in proof of His veracity and Divinity, and the subsequent foundation of the Church, proven by the miraculous illapse of the Holy Spirit of God - were not gathered into the community of disciples as the Lord intended, but for the moment (like Saul before his conversion) persecuted the nascent Christians and cast them forth from the synagogues. This split, once made official and definitive and permanent, marked all remaining rites now emptied of their objective worthiness, since their time had come to an end and, worse, they were being kept up in defiance of the will of the Deity, Who in wondrous manner had manifested His unimaginable plan of salvation in His Son.
Just as happened to the First Temple, in the terrible words of the annalists of Israel at the ends of the books of Kings and of Chronicles (or Paralipomenon), "the wrath of the Lord rose against His people, and there was no remedy" (II Par. xxxvi, 16) and Titus "burnt the House of the Lord... and the houses of Jerusalem" (IV Kings xxv, 9). Woe! The Lord's lament over Jerusalem came to pass with deplorable horrors and outrages committed.
But the mercies of the Lord are new every morning (cf. Lam. iii, 22f) - from Pentecost, down the first centuries, even until the present, many of Israel have heard and accepted that the Lord has done a new thing, a woman has within her without seed conceived a Man (cf. Jer. xxxi, 22), of Whom it is said, Thou art My Son (Ps. ii, 7). And we have the warrant of the Apostle that in the end all Israel shall be saved (Romans xi, 26 - the whole chapter bears careful reading). The Lord does not go back on His promises (cf. Rom. xi, 29): the Chosen remain such; and, wonderfully, the prophecies of old are fulfilled, that His mercy shall save even the Gentiles also, for in His mysterious Providence the sometime blindness and stubbornness of Israel helped provide a chance for the sinners of the Gentiles to be delivered too.
It is thus that the Fatherhood of God over all mankind is demonstrated, and His justice toward all the earth vindicated: His Son, breaking down the barrier between them, has made the twain one (cf. Eph. ii, 14).
For further reading, may I recommend St Thomas Aquinas in his Summa Theologiæ? He deals with the question of the duration of the ceremonial precepts of the Law in Ia-IIæ, q. ciii, articles 3 and 4.
Of your charity, dear commenter, do let me know if any of this is useful.