The early history of Ancient Rome is notoriously poorly documented. The Gaul Sack of Rome (390 BC) has destroyed much early written records. What evidence do we have that remains of the Roman Kingdom? I am no expert in Roman history, but is it even certain that the Roman Kingdom existed?
One theory is that the early kingdom period was actually a period of Etruscan domination which the Roman mythmakers (whose work is reflected in Livy) later reworked as the tale of the Tarquinian dynasty.
An interesting discussion of this can be found in the book The Beginnings of Rome: Italy and Rome from the Bronze Age to the Punic Wars by Tim Cornell. Chapter 6 is titled The Myth of 'Etruscan Rome'. Another source is here.
There is some reason to believe that the Lapis Niger includes a contemporary reference to the king, and it dates from the period associated with the monarchy. It could be argued that the use of 'rex' here is purely religious - just like in Greece the word continued to be used for religious purposes long after the political institution was left behind. However the unanimity of all the ancient sources on the existence of a monarchy at Roma counts for something, as does the longstanding (and very well documented) hostility towards monarchy as an institution in later Roman sources.
It's also worth mentioning that Polybius cites a treaty between Carthage and Rome which he claims dates from the first generation of the Republic; and he claims it was preserved in metal tablets in the offices of the aediles. This isn't totally impossible, the Pyrgi tablets are from almost the same exact date and are still legible.