And Roman succession was quite often (mostly?) by inheritance. The "five good emperors" were the big exception, but even they had a habit of adopting their successors.
I think it would be better to see it as a form of nepotism than monarchy in the cases of the Principate, especially considering the ideological considerations of adoption in Roman society. The Senate could (and in the case of Lucius Verus, did, or tried to until Marcus Aurelius threatened to resign) credibly refuse to appoint an Emperor to the position on the grounds that they did not approve of him. They have a very magisterial process of the exercise of Imperial power, even if it is, effectively, autocratic.
Yeah, it still doesn't read as a straight monarchy, exactly, which is a one part of why I think it's cool. My impression is that it got there eventually in the Byzantine period. Is that correct?
Yeah, they outright began using the term 'basileus' and even the dreaded Latin 'rex', with emphasis on Emperors who were biological children of the previous Emperors.
Still a lot of civil wars and coups though. No amount of monarchy can erase THAT particular Roman tradition.
I would note that the word Republic comes from Latin ('Res Publica'), and that the Romans regarded it as a unique form of government itself - one which concerned itself with the common good ('Res Publica' - 'a matter of the public') and not just the good of its decision-makers - something which belonged to the people, or at least to general society. Theoretically. Obviously there is a significant gap in execution. Also, that monarchy in other cultures of the period was extremely common.
With that in mind, I think the distinction is important, not just in the Roman conception of themselves, but also in the way we should view the justifications of the state apparatus.