Why has the Catholic Church adopted the use of Latin, the language of the crucifiers of Jesus?
Before going on I would simply like to point out two things.
First of all that the Language that Pontius Pilate communicated to Jesus during his trial was Greek. After all it was the language of commerce at that time throughout the Mediterranean world.
Secondly, let us not forget that Pilate's inscription on the Title of the Cross was written in Latin, Greek and Hebrew. "This title then read many of the Jews: for the place where Jesus was crucified was near to the city: and it was written in Hebrew, and Greek, and Latin" (John 19:20). Thus if one of the bystanders did not understand one language, he would hopefully be able to comprehend one of the other two languages.
Each of these languages are to this day considered a liturgical language: Latin for the Roman Rite of the Catholic Church; Konia Greek for the Greek Orthodox Church and Hebrew for the Jewish nation.
During the course of the first three or four centuries the sacred liturgy in Rome was Greek not Latin.
Ecclesiastical Latin differs from classical Latin especially by the introduction of new idioms and new words. (In syntax and literary method, Christian writers are not different from other contemporary writers.) These characteristic differences are due to the origin and purpose of ecclesiastical Latin. Originally the Roman people spoke the old tongue of Latium known as prisca latinitas. In the third century B.C. Ennius and a few other writers trained in the school of the Greeks undertook to enrich the language with Greek embellishments. This attempt was encouraged by the cultured classes in Rome, and it was to these classes that henceforth the poets, orators, historians, and literary coteries of Rome addressed themselves. Under the combined influence of this political and intellectual aristocracy was developed that classical Latin which has been preserved for us in greatest purity in the works of Caesar and of Cicero. The mass of the Roman populace in their native ruggedness remained aloof from this hellenizing influence and continued to speak the old tongue. Thus it came to pass that after the third century B.C. there existed side by side in Rome two languages, or rather two idioms: that of the literary circles or hellenists (sermo urbanus) and that of the illiterate (sermo vulgaris) and the more highly the former developed the greater grew the chasm between them. But in spite of all the efforts of the purists, the exigencies of daily life brought the writers of the cultured mode into continual touch with the uneducated populace, and constrained them to understand its speech and make it understand them in turn; so that they were obliged in conversation to employ words and expressions forming part of the vulgar tongue. Hence arose a third idiom, the sermo cotidianus, a medley of the two others, varying in the mixture of its ingredients with the various periods of time and the intelligence of those who used it.
Classical Latin did not long remain at the high level to which Cicero had raised it. The aristocracy, who alone spoke it, were decimated by proscription and civil war, and the families who rose in turn to social position were mainly of plebeian or foreign extraction, and in any case unaccustomed to the delicacy of the literary language. Thus the decadence of classical Latin began with the age of Augustus, and went on more rapidly as that age receded. As it forgot the classical distinction between the language of prose and that of poetry, literary Latin, spoken or written, began to borrow more and more freely from the popular speech. Now it was at this very time that the Church found herself called on to construct a Latin of her own and this in itself was one reason why her Latin should differ from the classical. There were two other reasons however: first of all the Gospel had to be spread by preaching, that is, by the spoken word moreover the heralds of the good tidings had to construct an idiom that would appeal, not alone to the literary classes, but to the whole people. Seeing that they sought to win the masses to the Faith, they had to come down to their level and employ a speech that was familiar to their listeners. St. Augustine says this very frankly to his hearers: "I often employ", he says, "words that are not Latin and I do so that you may understand me. Better that I should incur the blame of the grammarians than not be understood by the people" (In Psal. cxxxviii, 90). Strange though it may seem, it was not at Rome that the building up of ecclesiastical Latin began. Until the middle of the third century the Christian community at Rome was in the main a Greek speaking one. The Liturgy was celebrated in Greek, and the apologists and theologians wrote in Greek until the time of St. Hippolytus, who died in 235. It was much the same in Gaul at Lyons and at Vienne, at all events until after the days of St. Irenæus. In Africa, Greek was the chosen language of the clerics, to begin with, but Latin was the more familiar speech for the majority of the faithful, and it must have soon taken the lead in the Church, since Tertullian, who wrote some of his earlier works in Greek, ended by employing Latin only. And in this use he had been preceded by Pope Victor, who was also an African, and who, as St. Jerome assures, was the earliest Christian writer in the Latin language.
Development in the liturgy
Hardly had it been formed when church Latin had to undergo the shock of the invasion of the barbarians and the fall of the Empire of the West; it was a shock that gave the death-blow to literary Latin as well as to the Latin of everyday speech on which church Latin was waxing strong. Both underwent a series of changes that completely transformed them. Literary Latin became more and more debased; popular Latin evolved into the various Romance languages in the South, while in the North it gave way before the Germanic tongues. Church Latin alone lived, thanks to the religion of which it was the organ and with which its destinies were linked. True, it lost a portion of its sway; in popular preaching it gave way to the vernacular after the seventh century; but it could still claim the Liturgy and theology, and in these it served the purpose of a living language. In the liturgy ecclesiastical Latin shows its vitality by its fruitfulness. Africa is once more in the lead with St. Cyprian. Besides the singing of the Psalms and the readings in public from the Bible, which made up the main portion of the primitive liturgy and which we already know, it shows itself in set prayers in a love for rhythm, for well- balanced endings that were to remain for centuries during the Middle Ages the main characteristics of liturgical Latin. As the process of development went on, this love of harmony held sway over all prayers; they followed the rules of metre and prosody to begin with, but rhythmical cursus gained the upper-hand from the fourth to the seventh, and from the eleventh to the fifteenth, century. - Ecclesiastical Latin (Catholic Encyclopedia)
It is all too obvious that the Church's decision to make Latin the language of the Roman Rite had nothing to do with the fact that executioners of Jesus Christ were Romans and spoke Latin.