Hi Sean. The words "mysterium fidei" are, without a doubt, required for "the form" of consecration of the wine. Please don't take my word for it. I am nobody. Take the words of Pope St. Pius V in the Roman Missal. That missal was promulgated with these words from the Papal Bull Quo Primum:

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Furthermore, by these presents [this law], in virtue of Our Apostolic authority, We grant and concede in perpetuity that, for the chanting or reading of the Mass in any church whatsoever, this Missal is hereafter to be followed absolutely, without any scruple of conscience or fear of incurring any penalty, judgment, or censure, and may freely and lawfully be used. Nor are superiors, administrators, canons, chaplains, and other secular priests, or religious, of whatever title designated, obliged to celebrate the Mass otherwise than as enjoined by Us. We likewise declare and ordain that no one whosoever is forced or coerced to alter this Missal, and that this present docuмent cannot be revoked or modified, but remain always valid and retain its full force...
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Here is what is said in De defectibus, the instructions included in the Roman Missal:

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V - Defects of the form

20. Defects on the part of the form may arise if anything is missing from the complete wording required for the act of consecrating. Now the words of the Consecration, which are the form of this Sacrament, are:

HOC EST ENIM CORPUS MEUM, and HIC EST ENIM CALIX SANGUINIS MEI, NOVI ET AETERNI TESTAMENTI: MYSTERIUM FIDEI: QUI PRO VOBIS ET PRO MULTIS EFFUNDETUR IN REMISSIONEM PECCATORUM

If the priest were to shorten or change the form of the consecration of the Body and the Blood, so that in the change of wording the words did not mean the same thing, he would not be achieving a valid Sacrament. If, on the other hand, he were to add or take away anything which did not change the meaning, the Sacrament would be valid, but he would be committing a grave sin.
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Now, someone might say that the words "mysterium fidei" does not change "the meaning of the words." But St. Thomas Aquinas disagrees with that (https://aquinas.cc/la/en/~ST.III.Q78.A3) when he says:

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I answer that, There is a twofold opinion regarding this form. Some have maintained that the words "This is the chalice of My blood" alone belong to the substance of this form, but not those words which follow. Now this seems incorrect, because the words which follow them are determinations of the predicate, that is, of Christ’s blood. Consequently they belong to the integrity of the expression.

And on this account others say more accurately that all the words which follow are of the substance of the form down to the words, "As often as ye shall do this," which belong to the use of this sacrament, and consequently do not belong to the substance of the form. Hence it is that the priest pronounces all these words, under the same rite and manner, namely, holding the chalice in his hands. Moreover, in Luke 22:20, the words that follow are interposed with the preceding words: This is the chalice, the new testament in My blood.

Consequently it must be said that all the aforesaid words belong to the substance of the form; but that by the first words, "This is the chalice of My blood," the change of the wine into blood is denoted, as explained above (A. 2) in the form for the consecration of the bread; but by the words which come after is shown the power of the blood shed in the Passion, which power works in this sacrament, and is ordained for three purposes. First and principally for securing our eternal heritage, according to Heb. 10:19: Having confidence in the entering into the holies by the blood of Christ; and in order to denote this, we say, "of the New and Eternal Testament." Second, for justifying by grace, which is by faith according to Rom. 3:25, 26: Whom God hath proposed to be a propitiation, through faith in His blood . . . that He Himself may be just, and the justifier of him who is of the faith of Jesus Christ: and on this account we add, "The Mystery of Faith." Third, for removing sins which are the impediments to both of these things, according to Heb. 9:14: The blood of Christ . . . shall cleanse our conscience from dead works, that is, from sins; and on this account, we say, "which shall be shed for you and for many unto the forgiveness of sins."
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Therefore, St. Thomas assigns the words "mysterium fidei" to "the substance of the form," not to its accidents. This means the words that make up "the form" of the Sacrament of the Eucharist would not continue to be "the form" if the words "mysterium fidei" were removed. Those words would be something other than "the form" of the Sacrament of the Eucharist. Again, this is St. Thomas Aquinas, not me.

One might object that "the Church" can change things regarding the Sacraments, and later Popes were in their authority to remove the words "mysterium fidei" from "the form" of the Sacrament. But Pope Pius XII said otherwise in Sacramentum Ordinis when he said:

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"...the Church has no authority over the substance of the Sacraments, that is to say, in that which, as witnesses of the sources of divine revelation, Christ the Lord himself determined to be kept in the sacramental sign"
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And as Aquinas says, t is not by ecclessiastical law that "mysterium fidei" is included in "the form." He says those words were from Our Lord himself (https://aquinas.cc/la/en/~ST.III.Q78.A3.Rep9):

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Reply Obj. 9: The Evangelists did not intend to hand down the forms of the sacraments, which in the primitive Church had to be kept concealed, as Dionysius observes at the close of his book on the ecclesiastical hierarchy; their object was to write the story of Christ. Nevertheless nearly all these words can be culled from various passages of the Scriptures. Because the words, This is the chalice, are found in Luke 22:20, and 1 Cor. 11:25, while Matthew says in chapter 26:28: "This is My blood of the New Testament, which shall be shed for many unto the remission of sins." The words added, namely, "eternal" and "mystery of faith," were handed down to the Church by the apostles, who received them from our Lord, according to 1 Cor. 11:23: "I have received of the Lord that which also I delivered unto you."
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Finally, even if one still thinks that Paul VI could have changed "the substance of the form" (which he could not), he did not, in fact, intentionally remove the words "mysterium fidei" from "the form." In Missale Romanum, he simply said that those words were being moved from the middle of "the form" to the end of "the form."

The sacrilege was accomplished in the rubrics of the Novus Ordo Missal and in the GIRM by other Vatican officials. It is in the rubrics and in the GIRM that the priest is told that the "words of consecration" end BEFORE the words "mysterium fidei" are pronounced, and that being so, the rubrics require him to genuflect to an unconsecrated chalice. By this genuflection, the priest externally establishes his INTENTION to end the "words of consecration." He clearly shows that, by genuflecting and therefore adoring the chalice, that he believes that the "consecration of the wine" has been completed without any need to say the words "mysterium fidei."

So, I hope this evidence convinces you of the following:

1. The words "mysterium fidei" are required by Pope Pius V's Apostolic authority in promulgating the Roman missal.
2. Those words are theologically designated to be "the substance of the form" of the Sacrament by St. Thomas Aquinas.
3. That the "substance of the Sacraments" cannot be changed by the Church because they derive from Our Lord by witness of the Apostles, as taught by Pope Pius XII and Trent.
4. That the change that was officially promulgated by Paul VI to move the words "mysterium fidei" did not specify that those words would no longer be considered part of "the form." That false assumption was put into the GIRM and into the Novus Ordo rubrics. These bad instructions confused the priests who were taught to say the Novus Ordo with the belief that the words of consecration were finished before saying "mysterium fidei," thereby causing them to INTEND to consecrate the Eucharist using an invalid form.

The mysterium fidei is absent in the Ukrainian and Mozarabic rites (and probably others), and consequently, it cannot be essential for validity.

“The form for the consecration of the wine is this: Hic est enim calix Sanguinis Mei, novi et aeterni testamenti, mysterium fidei, qui pro vobis et pro multis effundetur in remissionem peccatorum. The words: Hic est (enim) calix Sanguinis Mei, are certainly essential.
“Certain authors agree that the other words: novi et aeterni testamenti,etc., also pertain to the essential form. St. Thomas himself seems to follow this opinion, although some theologians and other authors think that the Angelic Doctor felt quite otherwise. Whatever is thought of the opinion of the Sacred Doctor and of other theologians, the opposite view is the common opinion and is thus morally certain. In practice, he would certainly sin gravely who would omit these words, and if he had said the first words only, he ought to repeat the entire form conditionally.”

Dr. Kwak is the latest pro SSPX shill that begins with the SSPX position (begging the question) and then backfills the "reasons" for it.