Just heard on the history of Rome podcast about the sack of Rome destroying basically all the primary documentation. Welp.

The loss of the early priestly tabulae shouldn't have much relevance. In 390 there wasn't much of a city in the first place, and few, if any, public records existed besides the pontifical tabulae. Livy notes that these in particular were what were known to have been destroyed in the sack in 390:

etiam si quae in commentariis pontificum aliisque publicis privatisque erant monumentis, incensa urbe pleraeque interiere

Even if there was material in the pontifical commentarii and in other public and private memorials it was mostly destroyed when the city was burned.

Note that he mentions not written texts, but private and public memorials, "monumentis." There was little in the way of written text in the early city, and most of it--like the pontifical commentarii he mentions--was epigraphic. Monumenta are those things which serve to preserve a memory, which may include texts but more usually refers to monuments, memorials, inscriptions, and other more physical and imposing remembrances. These early monuments may--indeed, probably not unusually--have contained written inscriptions, such as that on the Lapis Niger, commemorating the events or people responsible for their erection. Certainly these commemorative inscriptions were extremely common by Livy's day. Any sort of public or private material, then, would have been in short supply to begin with, and was unlikely to have listed any sort of lengthy genealogies. The Roman state at no time kept official genealogical tables going back to the early Republic, and though individual families maintained such traditions these were private traditions whose authenticity could be thrown into doubt. Indeed, these genealogical traditions often were doubtful. For example, the gens Junia, a plebeian nobilis family, claimed to have originally been a patrician family that had been transferred to the plebeian order, despite the fact that from an extremely early date we see no patrician Junii. The claim was doubted even in antiquity--Plutarch knew of an alternate tradition that traced the descent of the gen Junia from a plebeian bearer of the name--and such claims are hard to substantiate even today.

So, then, what was lost? By far the major textual loss of the sack in 390 would have been the pontifical tabulae. The priestly colleges, and in particular the pontifex maximus posted records of urban events. These took the form of some kind of inscribed tablets, which were at least thought to be copied down onto some lengthy master copy. These inscribed tablets contained mainly information of religious significance: omens, vota, the establishment of new sanctuaries, harvests, eclipses, etc. Possibly they included military campaigns and other events which might involve religious activity or concern. They certainly did not contain genealogical information--the pontifical tabulae, like the priests themselves, were unconcerned with establishing the ancestry of the city's clans, except when it was necessary to prove status for religious purposes. It is unlikely in the extreme that any family could have pointed to any texts written before the sack in 390 to definitively prove descent from the Tarquins, even if such texts had survived (or, indeed, had existed in the first place).

To my knowledge there is no surviving family that claims descent from a Republican nobilis gens, nor were there generally in the Middle Ages. Mathematically speaking most of the human population should be descended from the Tarquins, though of course not all would be able to trace such a lineage or could claim direct descent through a male line. The Tarquins were part of a larger Etruscan population in the early city, and the expulsion of the ruling branch of the gens Tarquinius did not necessarily spell the death of the family in the city as a whole. Of the first consuls, one was a member of the gens Tarquinius: L. Tarquinius Collatinus, whose relationship to the ruling family was traced through Egerius, the son of Tarquinius Priscus' brother Arruns. Traditionally Collatinus' relation to the kings led to his abdication from consular office, and soon after the gens Tarquinia disappears from the politics of the Republic, as far as we know. But the gens Tarquinia did survive, probably through plebeian branches (as there is no confirmed existence of any patrician Tarquins after Collatinus). In 63 a certain Tarquinius, reportedly a follower of Catiline, was detained while trying to escape the city and join Catiline's army in Etruria. Tarquinius incriminated Crassus in the conspiracy, claiming that he had been sent by Crassus to urge Catiline to march into the city to free Lentulus, Cethegus, and the other Catilinarians whom Cicero had arrested--Tarquinius' testimony was considered unfaithful, due in no small part to Crassus' close cooperation with Cicero during the conspiracy, and Sallust reports that most people thought that either Publius Autronius (in order to disguise his own guilt) or Cicero himself had put Tarquinius up to it. Who this Tarquinius was exactly is unknown. It seems unlikely that he was anyone very important, and Sallust calls him "quidam L. Tarquinius", "a certain L. Tarquinius" as if his readers would have no idea who this guy was. At the very least the gens Tarquinia survived in name, although it's anyone's guess as to whether this gens had any actual ancestry from the royal Tarquins or whether they were simply another family that shared the name. Given that Tarquinii as a city was classified as a municipium in the historical period and that following 181 a colony was founded nearby it is not exactly hard to believe that a totally unrelated gens Tarquinia might have entered the city at a much later date. Whatever the origin of this Catilinarian Tarquinius, the name disappears at the end of the Republic--if any Tarquins still used the name by late antiquity they were not of enough importance even to appear in our epigraphy, much less our manuscripts.