There's multiple factors for why writings from Roman women generally don't survive:

The first is Roman education. Education in the Roman world was privatized, and both men and women went to school if they were from families wealthy enough to afford it. But Roman cultural practices typically had women come of age (and be married off) between the ages of 12 and 14, which resulted in the end of their schooling. As a result, upper-class Roman women never learned Greek, which was the language of the "Educated" that most upper-class Romans engaged in. And very few learned Latin overall, and their Latin education was cut short compared to their male counterparts.

Most Roman women were not major authors, with some exceptions (Anna Komnene, a Roman "princess" writing in the 12th century AD, being the big one, but that's outside the period in discussion.) However, many wealthy Roman women still wrote and engaged with each other through literature, but it survives through the writings of men or male copyists. For example, part of Aconia Fabia Paulina's poem she wrote as her own epitaph and had inscribed on her tombstone survives via the writings of the late Roman author Jerome. Jerome also mentions the poems of Cornificia, who lived around the time of Caesar and whose poems had survived until at least the late fourth century AD. Cornelia, the mother of Publius Cornelius Scipio Africanus, regularly engaged in writing letters to the members of her family, particularly her children, of which two excerpts survive via Cornelius Nepos. Her letters are also referenced by later authors. Other famous Roman female poets with surviving writings include Julia Balbilla (who was part of the royal family of Commagene, a client kingdom of the Empire in what's now Turkey and Syria), the 5th century Roman Empress Aelia Eudokia, as well as dozens of others either mentioned or who left little-known fragments. A nearly complete list can be found here.

Fundamentally, the issue is that Women's writings were largely not of interest to the early church fathers and monastic communities who copied classical and late Roman texts. The exception to this was early church women, as women played a MAJOR role in the development of early Christianity, as they played a major role in Roman funerary practices and death cults which early Christianity developed out of. Athough again, most of those writings are also lost, some do survive. Egeria, a Roman woman who embarked on a pilgrimage to Jerusalem between 381 and 386 AD, wrote multiple letters which were copied at Monte Cassino in the 11th century AD. Another example is the Passion of Saints Perpetua and Felicity, but the problem is that the text is heavily edited, so while the original was probably written by them, the version that survives today has been altered significantly by a male hand to enforce medieval themes about the roles and character of Christian women and the church heirarchy.

Interestingly in one of these cases, that of Faltonia Betitia Proba, she ordered her Pagan or Non-Christian poetry destroyed in her will, and only one of her Christian poems survives (however, it is basically a slightly modified copy of passages of Virgil reordered to be a life of Christ, and not exactly original). This really kind of sucks because she would be a huge source on the war between Constantius and Magnentius if she hadn't have ordered her non-Christian works destroyed.

Roman ("Byzantine") copyists in the middle ages, however, took a greater interest in classical literature. It's basically a common misconception that classical Greek and Latin literature survived thanks to the Arab world or the Catholic Church. While Christian scholars took an interest in a select few philosophers, and Arab scholars in some of the obscure scientific texts, the overwhelming majority of classical literature was preserved by the Romans themselves. Thankfully, this included many authors who preserved fragments of ancient Greek women's writings, such as the poetry of Melinno, who wrote in the tradition of Sappho, and fragments of which survive via Byzantine copies of Stobaeus and Athenaeus, who preserved her Ode to Roma (the personification of the city).

There are, however, at least six direct surviving examples of a woman's writing: the first is that of Claudia Severa in the Vindolanda Tablets. She was a Soldier's wife who inscribed four lines herself below those of a professional scribe. They read:

sperabo te soror

vale soror anima

mea ita valeam

karissima et have

The second is the poetry of multiple women who graffitied it into a monument on the left leg of one of the Colossi of Memnon. Unfortunately, I don't have a copy of the original Greek to quote, but the English translation of Caecilia Trebulla's poetry is provided on Wikipedia.

For further reading, I recommend Ian M. Plant's Women Writers of Ancient Greece and Rome: An Anthology.

I'm sure another flaired user who specializes in Women's writings and the transmission thereof can expound on my response. But I hope this answers your question.

We do actually have surviving writing by Roman women! The elegies of Sulpicia are probably the best known in Latin, the first of which I’ll throw in here (translation from the Loeb):

At last love has come; and the rumour that I have concealed it would shame me more than disclosure. Won over by my Muse’s prayers, Cythera’s queen has brought and placed him in my arms. What Venus promised she hath fulfilled. Let my joys be told by all of whom ‘tis said that they have missed their own. Never would I choose to entrust my messages to tablets under seal, that none might read them before my lover. Nay, I love my fault, and loathe to wear a mask for rumour. Let all hear that we have met, each worthy of the other.

The six elegies securely attributed to Sulpicia circulated in the third book of Tibullus, the so-called Appendix Tibullana, and for that reason they were initially attributed to Tibullus rather than Sulpicia.

(There is also another Sulpicia, whose surviving work consists entirely of a single quotation by a commentator on Juvenal, for which see Richlin 2014, 118)

We also have non-literary writing by women that survives as well. Check out a recent Tuesday Trivia comment of mine for a birthday invitation that survives from Roman Britain, and when I can I always love to share my hands-down favorite inscription from antiquity: a roof tile in Pietrabbondante. It preserves the side-by-side footprints and the bilingual Oscan-Latin inscription of two fellow slaves, at least and likely both of whom were women, probably working at a tile factory (transcription and translation from Wallace-Hadrill):

HN. SATTIIEIS DETFRI
SEGANATTED. PLAVTAD
(Detfri slave of Herennius Sattius/signed with a footprint)

Herenneis Amica
signavit qando
ponebamus tegila.
(Amica slave of Herennius/signed when/we were placing the tile).

Each language is in a different hand, so rather than one biliterate author this is the work of two literate, bilingual women creating something together - in fact each language provides a detail the other does not, making the collaborative whole important. It’s been most commented on by scholars interested in the bilingual nature of the inscription and its context, but it also shows literate women making their mark somewhere no one could see it.

In general though, you’re not entirely wrong. Generally speaking, the reason why is one of those a little of column A, a little of column B, a little of column C, where A, B, and C are:

A) The amount of literature we’ve lost from antiquity is in general enormous, and not only among women authors.

B) The dissipated nature of the corpus that does survive makes it difficult to see in the aggregate.

and

C) Scholarship is in general not very good at seeing women in the material record.

Let’s break those down.

A) The amount of literature we’ve lost from antiquity is enormous.

It’s quite difficult to overstate just how much we’ve lost from antiquity to the present - the generally-cited number is 1%, to give an idea - and the survival of even some of the texts we do have has been precarious, and transmission sometimes rests on the accidental survival of only one or two exemplars somewhere along the line or the chance finding of some papyrus somewhere with something on it. Against a general background of loss of texts, ancient women writers have certainly suffered disproportionately, but the unfortunate fact remains that for most texts survival has been the exception, not the rule.

B) The dissipated nature of the corpus that does survive makes it difficult to find.

In the documentary record, there are scattered texts by women - like the Pietrabbondante inscription, or Claudia Severa’s birthday invitation - just as there are in the literary corpus - that single, two line quotation of the imperial Sulpicia’s, or the hidden ‘Tibullan’ poems of the earlier Sulpicia - but often you have to already know what you’re looking for in order to find them. The Pietrabbondante inscription, for instance, has been commented on in its capacity as a bilingual text (Wallace-Hadrill, Adams), less so for its place in women’s history (it features briefly in Richlin, but the sum total of her comments on the text after a brief quote from Wallace-Hadrill are: “These two slave woman made a mark to show they were there, not only in words but in the flesh.”). Likewise the hidden lines of Sulpicia quoted by the Juvenal commentary aren’t exactly something one might simply stumble upon. While there are a number of source books about women’s lives, to my knowledge one of the only collections of solely women’s writing from antiquity is I.M. Plant’s Women Writers of Ancient Greece and Rome: An Anthology, which also includes an appendix of attested Greek and Roman women writers whose works we’ve lost. So even where we might actually have surviving writing, it’s not necessarily either obvious or accessible.

C) Scholarship is really bad at identifying women’s writing.

Last but certainly not least, scholarship is… really, really not great at identifying writing by women. To return to an earlier point, the elegies of Sulpicia that do survive were assumed to be by Tibullus because they circulated with his poems and because generally speaking until the author is proven a women they are generally assumed to be a man. I talked about this some in the birthday invitation post I referenced earlier in reference to identifying women’s handwriting in the material record, but much of the same applies to women’s authorship in the manuscript tradition. For anything of uncertain authorship, or pseudonymous, or anonymous, generally speaking the unquestioned assumption is that the author is a man unless definitive evidence proves otherwise, and as the case of Sulpicia proves, sometimes even then women’s authorship can be doubted.

Looks like u/FlavivsAetivs got in with an answer while I was working, so I’ll just throw out for the end here that while there were almost certainly more barriers for women than men in education, the existence of literature by women in antiquity means that we can say for sure some women did reach a sufficient level of education to engage with the literary tradition of their male peers on a highly literary level.

Women’s writing faced major barriers to survival, but it does survive.