The motu proprio Traditionis Custodes has reignited the debate over doing readings at the TLM exclusively in the vernacular.New Liturgical Movement: Against Vernacular Readings in the Traditional Mass
As most are aware, Summorum Pontificum had already opened up this possibility for low Masses, but it was seldom used, since people tend to understand that the readings are an integral part of the missal and of the act of worship, and that a continuity ought to be kept among all elements of the liturgical action. In addition, readings were already being given in the vernacular from the pulpit prior to the homily, and most of the faithful have translations in their hand missals. By and large, it is a non-question and a non-starter within the TLM world, and the latest assault on the integrity of the Latin liturgical tradition has met with principled and pragmatic resistance (see here and here).
Nevertheless, this question deserves to be revisited from time to time in order to understand better the rationale for sticking with the tradition. Here’s what a friend who is quite sympathetic to the TLM wrote to me:
I personally find that one of the best things about the Novus Ordo is vernacular readings. I take a via media approach; I don’t believe “pastoral” adaptions should be made in the liturgy, but I do enjoy how, in the Novus Ordo, the Word is proclaimed in the vernacular. When I have been at Latin Masses, I love the chanting of the Epistle & Gospel, but then when the priest goes and reads it from the pulpit before the homily, it is often done in a rushed, sloppy, and awkward manner. Perhaps you have written a response to my objection, and I missed it. What is the justification for retaining the readings in chanted or spoken Latin? Like I said, I think it’s beautiful, but in my idealized liturgy which I imagine to be the fruit of a Third Vatican Council called for by Cardinal Sarah-turned-Pope Benedict XVII, it largely looks like the 1962 Missal but with vernacular readings.
This is indeed a complex question. There are two aspects of the issue. First, what is the purpose of the reading of Scripture at Mass? And second, how can we practically overcome the language barrier that Latin presents to most?
In terms of the first aspect, there is no doubt that the traditional liturgy understands everything as doxological and latreutic. Nothing is merely didactic or informative. (In fact, this is why the homily strikes us as an interruption in the action: it is certainly merely didactic and informative, and therefore doesn’t smoothly harmonize with the rest of the liturgy, which is a ritual, a sacred action.) Because of this orientation to God, the readings are chanted like prayers, incense is used, a ceremonial with a procession is followed. The Novus Ordo was unfortunately composed at a time when it was all the rage to think of readings at Mass as a sort of communal Bible study, and that is why the Liturgy of the Word is so dreadfully verbose and static. Everything is read (almost never sung), towards the people, from the ambo, and without a sense that this Word is being offered up to God and raising the minds of the faithful up to Him in prayer. 
At the Latin Mass, everything is done for God as well as for the people: nothing is “just for the people,” as if we’re turning our backs on God and saying: “Pardon us, we have some business to take care of now; we’ll come back to You later.” The classical phrases used to describe the two main parts of the Mass— “Mass of the Catechumens” and “Mass of the Faithful”—each speaks of a missa, and this, not only because there are certain categories of people “sent away” (first, the catechumens, and then the faithful), but also because, as the medieval commentators explain, missa est means “it is sent”: our offering to God is sent up to Him by the hands of angels! In ancient Israel as in the Church, much of our worship consists in offering words up to God as a verbal sacrifice, parallel to our offering up of incense to Him. As incense pervades the church but also rises up, so too does the Word of God: it is not shot forth to the people (as if they are the pupils drilled by a teacher), but exalted so that it may rain down on them. Yes, there is something sacramental and mystical in this descent: there is a blessing in the repetition of the hallowed words of the liturgy that goes beyond its rational content. In the Liber specialis gratiae, St. Mechtild of Hackeborn says that Christ spoke to her these remarkable words:
You shall understand that when you say any psalm or prayer which any saints prayed when they were alive on earth, then all of those saints pray to me for you. Additionally, when you are in your devotions and speak with me, then all of the saints are joyful and worship and thank me.
It is surely no small thing for us to be reciting and singing the very same words that most of the saints of the Latin or Western half of the Church have had on their lips across all the centuries. These are words of diachronic unity, reverberating harmony, and revelatory power.
In terms of the second aspect mentioned above, it seems there are better ways to accomplish the good of comprehension than chucking out a stable practice of over 1,600 years’ duration and replacing it with the use of embattled and prosaic compromise translations that please no one, being (depending on who you are talking to) dated, too casual or too formal, too loose or too literal, etc. Most modern Westerners are still literate enough to find following along in a missal no difficulty, and since the translations in the missals are not official, they can vary in style. I have come to prefer this multi-sensory and more laissez-faire approach. If the reading from the pulpit is done well, it reinforces the proclamation. On most Sundays I engage with the reading multiple times: at Mass when I hear it in Latin and possibly read it; again when it’s read from the pulpit; and then in the parts that come up in Vespers. The old approach in fact saturates you slowly in Scripture rather than spraying you with it in great buckets.
We can and should also make a concerted effort to be teaching Latin to all Catholics, children and adults. Any serious religion teachers serious stuff to its followers: the Jews teach Hebrew, the Moslems teach classical Arabic, etc. If we cared about our heritage, you can bet that every schoolchild would be translating passages from the Vulgate, which is a more enormously consequential text in the history of the West than Homer, Dante, Shakespeare, or [insert favorite famous name].
I deliberately place the next consideration after the foregoing points because I do not wish to be accused of aestheticism. However, it is quite true, and rather obvious, that the Tridentine liturgy possesses a colossal unity of form and substance, a unity to which the use of Latin makes a significant contribution. I’m reminded of a passage from Samuel Johnson, commenting on an epitaph he saw that was half in English, half in Latin:
It may be proper here to remark the absurdity of joining in the same inscription Latin and English or verse and prose. If either language be preferable to the other, let that only be used; for no reason can be given why part of the information should be given in one tongue, and part in another on a tomb, more than in any other place, or any other occasion; and to tell all that can be conveniently told in verse, and then to call in the help of prose, has always the appearance of a very artless expedient, or of an attempt unaccomplished. Such an epitaph resembles the conversation of a foreigner, who tells part of his meaning by words, and conveys part by signs.
At the same time, the liturgy, which is too great for any one of us to say he “understands” in full with no reminder, can legitimately be compared to Johnson’s “foreigner who tells part of his meaning by words, and conveys part by signs.” For the words alone are not enough, nor are the non-verbal signs, but together they constitute a whole that is greater than its parts. We understand the uniqueness and the divine authorship of the words of Scripture better when we hear them read or chanted in Latin than if we heard them only in the vernacular; but their exalted status is no less emphasized by the treatment of the book, the kissing of it, the incensing of it, the processing with it. We don’t do that kind of thing to ordinary books.
It is often said that a major driving force in the Catholic liturgical reform was the secret Protestantizing sympathies of many of the liturgists and their not-so-secret obsession with lowest-common-denominator ecumenism. That seems to be true in all kinds of ways. We should not forget, all the same, that most of the early Protestants were a good deal more conservative, more “traditional” in their instincts, than the Catholic liturgists of the 1960s or their ragtag sympathizers today. I wrote about this elsewhere in connection with the manner of receiving Holy Communion, but here is a quotation from Martin Luther talking about his desire to preserve the ancient languages in worship:
Now there are three different kinds of Divine Service. The first, in Latin, which we published lately, called the Formula Missae. This I do not want to have set aside or changed; but, as we have hitherto kept it, so should we be still free to use it where and when we please, or as occasion requires. I do not want in any way to let the Latin tongue disappear out of Divine Service; for I am so deeply concerned for the young. If it lay in my power, and the Greek and Hebrew tongues were as familiar to us as the Latin, and possessed as great a store of fine music and song as the Latin does, Mass should be held and there should be singing and reading, on alternate Sundays, in all four languages—German, Latin, Greek, and Hebrew. I am by no means of one mind with those who set all their store by one language [in context, this seems to mean German].
Of course, I wouldn’t say we should do anything, or keep something, because Luther said so or did so. Rather, the point is that the “Catholic” liturgical reformers and implementers—including Paul VI—were, in certain ways, more Lutheran than Luther himself. That’s why the pope’s good friend Jean Guitton was right to say in an interview that Paul VI’s intention was “to get the Catholic Mass closer to the Calvinist Mass [sic].”
Meanwhile, the truly universal or catholic, and dare I say Pentecostal, attitude of the Catholic Church was well expressed by Maisie Ward in 1937, in sentiments that have been echoed and reechoed by countless laymen and clergy down through the centuries:
This union of localization and universality finds expression in the miracle of tongues on Whit Sunday and to-day in the language and liturgy which unites, at one altar, men severed by national languages and national interests.
 I recommend the FIUV position paper on this subject, which packs a lot into a few pages.
 On the “Epitaph to James Craggs,” from Johnson’s Life of Pope.
Documents Illustrative of the Continental Reformation, ed. B. J. Kidd (Oxford: Clarendon Press, 1911), 195.
The Wilfrid Wards and the Transition, vol. 2: Insurrection versus Resurrection (New York: Sheed & Ward, 1937), 7.