The way the Roman 'republic' was doing it was an unelected administrative state, so by your way of thinking, there was no real break between not having an emperor and having one.
From Wikioedia...
“With MODERN republicanism, it has become the opposing form of government to a monarchy and therefore a modern republic has no monarch as head of state.”
So the differentiation between a republic and a monarchy is of recent origin (comparatively speaking).
Back in Roman time what they called a republic was definitely not what we think of a republic today. There was no voting by the public in general.
In fact even in the US, right after independence only land-owning white males could vote.
Just think how many times the voting rules have changed just within the short history of the US. And they continue to change - not for the better I might add.
But wait, there is more. Early roman emperors were elected.
“The legitimacy of an emperor’s rule depended on his control of the army and recognition by the Senate; an emperor would normally be proclaimed by his troops, or invested with imperial titles by the Senate, or both.”