You're confusing here the oldest written Latin literature and the oldest surviving Latin literature.

In truth, it is rather impossible to try to fix a date to when Romans started writing down literature. The earliest written Latin survives from as early as the 7th century, as graffiti on crafted artifacts such as the Praeneste Fibula. Interesting about these sorts of objects is that finding 'X made me' - maker 'stamps' suggests that literacy wasn't even limited to the upper classes in the very earliest periods of written Latin, if an lowly artisan could write. It would be weird that graffiti was the only literary genre in existence. Most written archaic Latin was mainly written on non-durable materials, which is why only graffiti and few stone inscriptions survive from these early days, so what is left for academics is to try to untangle the massive melting pot of cultures, languages, and literary forms that is Republican Roman literature. It has been obviously influenced by so many cultural currents that it is rather difficult to pick out the clues that betray Latin literary traditions older than Latin drama, but it is possible. It is rather clear that Latin had been used for hundreds of years before Plautus to record things important for public, private, and religious functions: preservation of oracles, religious formulas and prescriptions, lists of magistrates and priests, statutes, laws, treaties, letters, family genealogies, and so on. There are a number of literary 'forms' that we know to stretch far back to the history of Latin literature, beyond Plautus et co., that haven't survived to our day. Here's a couple of the most important ones, and an explanation on how we have evidence of them:

1. Poetry, especially religious poetry, spells, prayers and such were written down and recorded. One example and the (as is assumed) oldest piece of surviving Latin literature might be the Carmen Saliare, fragments of a religious chant of the Salian priests, preserved in a number of later authors (eg. Hor. Epist. 2. 1. 85–6; Quint. Inst. 1. 6. 40). They are written in Archaic Latin and believed to be datable to perhaps as early as 700 B.C. The text and interpretation are much more disputed than the Wikipedia article makes clear, and the exact form of the poem was already corrupted in antiquity (the archaic Latin was probably just as unintelligible to Republican Romans than it is to us), but it is clear that poetry was created and written down for religious purposes very early in the history of Latin. It is of course possible that an epic tradition of oral heroic poetry, comparable to Homeric poetry, existed in Latium way before first written Latin epics, but this is pure speculation.

2. Street theatre and the so-called Fescennine verses - these sort of forms of 'oral literature' hasn't been really passed down on us, but just as we have surviving fragments of fabulae praetextae and later Roman mime (e.g. low-brow, satirical, obscene-natured popular street theatre) it seems likely that these were also written down and recorded way before Plautus. Catullus and Horace among others talk of these traditions, which were surely deeply ingrained into the Roman society when the historical period starts, and were probably an influence to Plautine comedy and Roman satire. The Romans certainly believed that the traditions were ancient, and that the so-called Fescennine verses were called so either because they originated from Fescennia, an Erutrian town, or from the word fescennium, which means both 'evil eye' and 'penis' (I know, I know...). Fescenni were oral, improvised pieces of poetry that however had stock phrases and characters, of mocking and comical characters, which might even become a form of public defamation. They were especially enjoyed at rustic festivals, and they might have even had an apotropaic function, that is, keeping the evil eye at a distance.

3. The Annales and the Fasti. This is another very ancient form of writing: the Roman society had developed a public calendar (the Fasti), which was regulated and sanctioned by the religious authorities. The pontiffs had to e.g. divide the days of the year to fasti and nefasti; on the days that were nefasti, it was not permitted to conduct public affairs for religious reasons. This information needed to be spread and made public, and made a record of for future purposes (i.e. for the interpretation of the coming year, it might be important to see, what days had been 'unlucky' in the past). Over time, more information started to be written into the fasti, and this developed into the so-called annales: yearly day-by-day records written by magistrates, that included information such as official religious acts, names of the magistrates for the current year, events of public concern, treaties, proclamations of wars, natural disasters, prodigies, and so on. The annales formed a collective memory of the Roman state. When Publius Mucius Scaevola, in the time of Gracchi, c. 130BC, collected together all of the annales, they stretched back 280 years. The annales were a majorly important source to the Republican and Imperial Roman historians. The annales cannot maybe be considered literature, as it doesn't have a coherent narrative voice, but it's probably worth mentioning here.

So, I don't know if we can say that the Latin literature culture did not begin until the 3rd century - although this is what the Romans believed. They (eg. Livy 27.37.7 and Cicero Brut. 18.72-74) actually had a fixed date for the 'birth' of artistic Latin literature: 240 B.C., the year when a Greek freedman Livius Andronicus, an expert of Greek drama, put on the first original play ever in Latin. Previous to his play Andronicus is known to have translated the Odyssey into Latin. Andronicus is the first known Latin literary figure, for sure - he pre-dates Plautus by some decades, but only few scattered lines of his Odyssey translation survive.

Obviously, after Andronicus, Latin literature explodes, and we get Plautus, Gnaeus Navius, Ennius, Terence... and probably a number of other artists and drawer poets that are lost to us. So, it is sort of tempting to see Andronicus as the father of artistic Latin literature, though most likely this isn't exactly the case, as I've just shown how we have lost centuries of Latin literary genius that pre-date him. What Andronicus certainly was is the first product of the distinct Latin literary culture, which was imported from the Greek world - that is, the circulation of artistic works, poetry readings, an elite education based on artistic literature and rhetoric etc... This sort of culture was either unknown, not popular, or not widespread enough, so that works of Latin literature - that certainly or almost certainly were produced (Heroic poetry? Family histories? Religious mythologies? Comic street satire?) and almost as likely written down in some forms - would have been recorded down in durable material or circulated in any wider context than within small communities, and thus records and knowledge of them has been lost. The hundred years between the war against Tarentum (280-272 B.C.) - which is when the Romans were first heavily exposed to the Greek culture - and the invasion of Greece saw the Romans becoming rapidly Hellenized. During this time, not only did the Romans become infatuated with Greek literary culture, they also were sort of jealous of the Greeks' own characteristic forms of literature and language, and the Romans were keen to create a distinctly Latin literary culture and heritage. This is why we get Andronicus, Plautus and co. when we do - in the second half of the 3rd century B.C.

So, this probably my best answer to the OP's question: artistic Latin literature was produced, but it wasn't valued or circulated like Greek literature so that it would have taken comparable grand narrative scales and forms of mass-circulation, until the Romans learnt the 'Greek way'. I don't really have any explanation why the Romans didn't develop one independently - a simplistic explanation might be that archaic and early Republican Rome was a rather militant culture, and perhaps learning and education simply weren't valued independently of religious or practical aims. But this is obviously purely speculative and probably too 'neat' to be the whole picture.

Sources include ancient authors, mainly Livy, Cicero, Catullus, Horace; G. Conte and J. Solder's (1999) Latin Literature: a History; J.S. Harrison's (2005) A Companion to Latin Literature; Dr Costas Panayotakis' lectures.

Edit: spelling

I don't have a good answer, but rather some questions about your question, if you'll put up with them.

First, how do you define "literature"? Some Romans - Cicero, for instance - excluded annals and commentarii from the category of "history" because history is a literary genre (in his opinion) and those others are not. Roman law codes - particularly the Twelve Tables - seem to be quite old, as well as some of the ritual formulas preserved in Livy. We don't typically consider these literature, but is that a fair judgment?

Second, what about stuff we know existed and is lost? The almost entirely lost genre of Saturnian verse? The fabulae praetextae (since you mentioned plays) predate the surviving Roman comedies even though sometimes only a fragment of a sentence is left.

And what about oral literature that was never rendered into text and has been entirely lost?

Third, is it really a late point to expect the development of Latin Literature (given the understanding we seem to be operating under)? How often does literature develop in a total vacuum? Is it really surprising that Romans start writing history, generally considered a Greek genre in origin among us Eurocentric crowds, only when Greeks started writing history about Romans? Or adopted stage forms - again, generally considered Greek - when in close contact with Greek culture?

This point is fairly open to the criticism that Greeks didn't really invent these things, but my point here is to ask how often such things develop in a vacuum - without contact with a history-writing play-producing culture, would we expect a second culture to spontaneously develop those genres, never mind the question of getting around to writing that stuff down?

Once we nail down what literature is, how to deal with stuff that's lost, and what the expected arc of a literary culture looks like, we can start talking about whether Romans were late bloomers and if so for what reasons.