It's important because her beauty and miraculous, virginal childbirth manifest God's omnipotence. Her bodily integrity is a sign that she is sinless, immaculate.

Bodily beauty and integrity

Not just one of her body parts (her hymen), but her entire body (Merkelbach, O.P., Mariologia pp. 215)

113. […] always remained integral: immune from all corruption, especially illness, affliction, infirmity resulting from dishonorable and infamous sins arising from the rebellion of the organs, and from all blemish. This integrity can generally be called incorruption; and it can be viewed in particular in the end or immortal state of glory, or in the present and mortal life, in the wayfaring state: perpetual virginity especially serves integrity.

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^{§113. […] perfectum mansit semper integrum : immune ab omni corruptione, præsertim morbi, ægritudinis, infirmitatis ex sequela inhonoranti aut infamanti peccati ortæ et organorum rebellione, et ab omni maculatione. Hæc integritas generali nomine vocari potest incorruptio; at speciatim spectari potest, tum post hanc vitam, in statu termini seu immortalis gloriæ, tum in vita præsenti et mortali, in statu viæ : integritas viæ præcipue est perpetuo servata virginitas.}

It is important because her body was very beautiful (ibid. p. 214):

112. The body of Blessed Virgin was of a perfect complexion and beautiful.

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^{112. Corpori B. Virginis perfecta erat complexio ac pulchritudo.}

She didn't experience bodily pain (not even in childbirth), though she could be and was at times sorrowful (sorrow being of the intellect, pain being bodily; Our Lady of Sorrows is one of her titles).

Virginity in childbirth

One manifestation of the integrity of her body is her virginitas in partu (that she remained an inviolate virgin during parturition). This de fide dogma, an article of faith, is attested by Scripture (e.g., Song 4:12, Ezechiel 42:2) and the Fathers of the Church. The Catechism of the Council of Trent, describing Article 3 of the Creed (that Jesus was "born of the Virgin Mary"), says:

The nativity of Christ transcends the order of nature

But as the Conception itself transcends the order of nature, so also the birth of Our Lord presents to our contemplation nothing but what is divine.

Besides, what is admirable beyond the power of thoughts or words to express, He is born of His Mother without any diminution of her maternal virginity, just as He afterwards went forth from the sepulchre while it was closed and sealed, and entered the room in which His disciples were assembled, the doors being shut (John 20:19); or, not to depart from everyday examples, just as the rays of the sun penetrate without breaking or injuring in the least the solid substance of glass, so after a like but more exalted manner did Jesus Christ come forth from His Mother’s womb without injury to her maternal virginity. This immaculate and perpetual virginity forms, therefore, the just theme of our eulogy. Such was the work of the Holy Ghost, who at the Conception and birth of the Son so favored the Virgin Mother as to impart to her fecundity while preserving inviolate her perpetual virginity. [On the Nativity of Christ see Summa Theol. III a. 35, a. 36.]

3 types of virginity

There are three types of virginity (Ott, Fundamentals of Catholic Dogma bk. 3, pt. 3, ch. 2, §5):

1. "virginitas mentis [of the mind], that is, a constant virginal disposition"
2. "virginitas sensus [of the sense], that is, freedom from inordinate motions of sexual desire"
3. "virginitas corporis, that is, physical integrity."

"The Church doctrine refers primarily to Her bodily integrity."

Relation between "virginity of the body" and "virginity of the mind"

Even if St. Lucy was violated and conceived (Summa Theologica suppl. q. 96 a. 5 ad 4),

she would not have for that reason forfeit[ed] her virginity: nor would she be equal to Christ's mother, in whom there was integrity of the flesh together with integrity of the mind.

St. Thomas's explanation for why the Church doesn't consecrate violated virgins helps explain why an intact virgin is a more perfect virgin than a violated one (Super Sent. lib. 4 d. 38 q. 1 a. 5 ad 4; transl. Stegman p. 53):

Those women who are corrupted through violence, if they do not consent in any way, do not lose the glory of virginity in the sight of God. But since it is extremely difficult that in such pleasure some impulse of enjoyment does not surge forth, therefore the Church, which cannot judge interior matters, does not veil a woman among virgins when she has been externally corrupted. For which reason Pope Leo says: Those servants of God who have lost the integrity of their chastity by barbaric oppression will be more greatly praiseworthy in their humility and modesty, if they do not dare to unite themselves with uncontaminated virgins.

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^{illæ quæ per violentiam corrumpuntur, si nullo modo consentiant, virginitatis gloriam quo ad Deum non perdunt. Sed quia valde est difficile quod in tali delectatione aliquis placentiæ motus non insurgat, ideo Ecclesia quæ de interioribus judicare non potest, cum exterius corrupta sit, eam inter virgines non velat; unde Leo Papa dicit: illæ famulæ Dei quæ integritatem pudoris oppressione barbarica perdiderunt, laudabiliores erunt in humilitate et verecundia, si se incontaminatis non audeant copulare virginibus.}

Virginity consists in perpetually vowing to forgo such pleasure, irrespective of whether that pleasure actually arises in in a partuclar act of violent rape.

Discussing whether God can restore lost virginity, St. Thomas writes (Quodlibet V q. 2 a. 1 co.; English transl.):

There are two things to consider about virginity. One is the integrity of mind and body. In that respect, God can restore a virgin after falling. For God can renew the mind by grace and repair the body by a miracle. The other thing to consider is the reason for that integrity, viz. the fact that a virgin woman has not been known by a man. In that respect, God cannot restore a virgin after falling.

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^{in virginitate duo possumus considerare. Quorum unum est ipsa integritas mentis et corporis; et sic Deus virginem potest reparare post ruinam: potest enim mentem reintegrare per gratiam, et corpus consolidare per miraculum. Alia autem est causa integritatis praedictæ, quia scilicet mulier virgo non fuisset cognita a viro; et quantum ad hoc Deus non potest virginem post ruinam reparare}

A non-violated virgin is greater because she has "has not been known by a man".

Artistic portrayals

• Jaki, Stanley L., O.S.B, “The Virgin Birth and the Birth of Science,” The Downside Review 107, no. 369 (October 1, 1989): 255–73.

cites William Blake's Nativity

as "one of the greatest but possibly least appreciated treasures of the Philadelphia Museum of Art" (p. 266), saying that "Blake painted that Nativity as if guided by" "these words of Jerome" (p. 267), from Perpetual Virginity of Blessed Mary: Against Helvidius c. 4 (Jaki's translation of Migne PL 23:201):

Should the woman giving birth be overtaken by pain, midwives pick up the crying infant and the husband will hold the exhausted wife. … But in no way should this be thought of the Saviour's mother and of that just man, Joseph. Here is no midwife; no need here for women to be fussing about. His mother herself wrapped him in the swaddling clothes, herself mother and midwife.

... why is it so important for Mary's bodily integrity to remain intact as regards her "perpetual" virginity during childbirth?

Many of those who, like me, were educated in the Catholic Church, have always felt uneasy about the teaching of the triple virginitas (ante partum, in partu and post partum).

To affirm that the Virgin Mary didn't undergo a rupture of her hymen during childbirth means to affirm that not only her conception was miraculous (which is clearly affirmed in the NT, in particular Matthew 1:18 and Luke 1:35), but even that childbirth was miraculous, which is nowhere affirmed in the NT.

Besides, the other possible explanation of the virginitas in partu is Docetism, which turns the humanity of Jesus into delusional appearance.

One of the firmest opposers of the doctrine of the virginitas in partu was Tretullian, in his On the Flesh of Christ (transl. Evans, 1956).

Conclusion

The doctrine of the "perpetual" virginity during childbirth is very confusing for the faithful, because it goes against the real humanity of Jesus.

The answer is simply at that time that was the sole measure of virginity, regardless of how some came to think about it centuries later. Basically, it is important because it was the proof of virginity.

This is shown in scripture and in a very early tradition.

And give occasions of speech against her, and bring up an evil name upon her, and say, I took this woman, and when I came to her, I found her not a maid: Then shall the father of the damsel, and her mother, take and bring forth the tokens of the damsel's virginity unto the elders of the city in the gate: Deut 22:14-15

The tokens of virginity included blood and breakage.

We find this same integrity idea in the Infancy Gospel of James.

[19]Then said Salome: As the Lord my God liveth, unless I thrust in my finger, and search the parts, I will not believe that a virgin has brought forth. 20. And the midwife went in, and said to Mary: Show thyself; for no small controversy has arisen about thee. And Salome put in her finger, and cried out, and said: Woe is me for mine iniquity and mine unbelief, because I have tempted the living God; and, behold, my hand is dropping off as if burned with fire. source

This absolutely necessary physical integrity found its way into the Catholic Dogma (de fide).

57. This union of the mother with the Son in the work of salvation is made manifest from the time of Christ's virginal conception up to his death; first when Mary, arising in haste to go to visit Elizabeth, is greeted by her as blessed because of her belief in the promise of salvation and the precursor leaped with joy in the womb of his mother (cf. Lk. 1:41-45); then also at the birth of Our Lord, who did not diminish his mother's virginal integrity but sanctified it, source

Conceived without blood or breakage. Birthed without blood or breakage. Mary's physical virginity remained intact as proof she was and remained a virgin they believe.