I'm not sure that if Rome kept most of its earlier archaic features, you'd have the social and political conditions to the IOTL expansion : at the contrary, it could be considered as part of the Etruscean commonwealth rather than something much distinct.
That said, it doesn't mean it wouldn't evolve towards a more ceremonial kingship.
Indeed, archaic kingship didn't really managed well to stick with city-states and generally did so by becoming a mostly ceremonial magistrature.
One could see a parallel between the dual magistrature in Rome and the dual kingship in Sparta as for matter a demos-based kingship (contrary to ethnic kingship as in Macedonia), but the latter case may be accidental and coming from three original spartian tribes. (Rome is supposed to have hosted three original tribes as well, but it's hard seeing what's part of a more or less genuine historicity, and what could come from the IE tri-functionality* as much as genuine organisation).
A non-etruscean ethnic kingship could have less troubles maintain itself in Rome as it did in Sparta : however, this would ask a really dark (knowledge-wise speaking) and butterflying PoD, when a "civic" kingship would be less problematic on this regard for what matter immediate outcome, but may be less stable.
Rome, as an independent city, could less tolerate attempts againt its regional hegemony over Latins from the powerful Etrusceans that, while far from being united, represented a political threat as in possibly limitating Roman power or even structures, making the old aristocracy entering a commonwealth that limited their chances of political and social gains.
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Assuming we're talking of an then latino-etruscean kingship, we could see the rex evolving towards a more and more religious role, while praetori assuming the day-to-day management, and ending eventually with a rex elected the same way than the IOTL interreges and flamines.
This would change little structurally, while it could mean a more stable Roman state in its archaic history, basically a "crowned republic".
It could mean a more conservative, religiously speaking, Romanity, and IMO less bound to favour military expansionism.
*respectivaly Ramnes, Luceres and Tites or first, second and third.