Basileus was not felt as an equivalent to the Roman "Rex". Critically, the Greek word had extensive precedent for usage in a sense akin to a republican magistrature and/or collective leadership, the Spartan diarchy being the foremost but not unique example. This does not mean, of course, that actual Roman Basileis were fine with sharing power or conceived their rulership as anything less than singular and absolute, (not in normal circumstances at least) but the theoretical notion that they were ultimately, in a sense, magistrates was ideally consistent with their titles.* "Rex" did not have the same load.
Which is particularly funny in hindsight, given how republican the office of Rex was in Early Rome. The Senate had to recognise the king, the succession was not strict primogeniture, the King actually had to get his heir recognized as well. The main pitfall I believe of the early kings is just that they somehow became foreigners. (I actually want to know why suddenly the Kings started being Etruscan, was it the result of military defeat?)
Compare the codified succession of Roman kings to the free-for-all that occurred in the Empire and it just breathes the air of the Cromwell situation. Some guy takes power, refuses the crown and its set of powers for undefined unconstitutional power and equally undefined succession that ultimately becomes worse for the state than any Rex would have been.