I haven't read the book yet, though I am interested to do so. In the History of Byzantium podcast, Kaldelis is interviewed about his book, for about an hour, where he explains his views. That is, that the Byzantine society and system was less divine monarchy than historiography often potrays, and that the people of Constantinople played often a key role in legitimizing emperors or the change in power.
That there was some ideological continuity from the republic, to the empire, and later to the eastern roman empire/Byzantine empire regarding the concept of res publica. Which is often translated as state, but in reality it speaks more of the concept of commonwealth, as it was understood by the Romans.
I think Kaldellis' book is a wonderful read and that his argument will have a big impact in the future. However, as I noted here, Kaldellis did not cover in enough detail about all the changes and exceptions to this theory (admittedly it's a fairly short book), and he did not acknowledge that existing studies of Byzantine government were already moving towards a more nuanced view of the monarchy, though he is correct to note that we have to think outside of the box more. I've only been able to find one academic review so far and it is sadly quite negative, but who knows if it is representative of Byzantinists' views in general. Anecdotally, I also have to note that his other ideas about various famous Byzantine historians being crypto-pagans are somewhat unpopular amongst the academics I've spoken to.
There is a lot of useful material in this book though and I think he is absolutely right to demand historians to think again about what we know of the Byzantine state. As his book was only published earlier this year, I think we have to wait a few years before more reactions are published and other historians can engage more fully in his work. It is far too early for there to be a consensus on this, though personally I'm a fan of his ideas, if not of the extent he pushed them.
I've been pondering over the premise of this book for some time, and it's certainly going to play some role in my eventual thesis. Certainly I'm warming to his opinions, but only because, as you said, we need to think of the Res Publica as something different from our english word for "republic." The Greek word Politea is probably a better fit.
I think Kaldellis talks about this in his book, but I've also been thinking that the idea of a broader "Roman political community", be it Byzantine or high Roman, where the populace has an impact in the power system of the empire is really only due to the fact that the important centers of power are physically manifested in a centralized location. In the case of the high empire it's the city of Rome, in Byzantium it's Constantinople.
In the intermediary period of Diocletianic imperial mobility, the people as well as the lower level bureaucratic apparatus had less a part to play because the emperor was frequently away from fixed locations where they could play a decisive factor as a threat to his power (e.g. the Nika riots). One could argue that this was in fact, by Diocletian's design, although it was mostly due to the necessity of keeping an emperor mobile to respond to constant 3rd century style threats.
Once this mobility stopped in the 5th century for the east, you begin a reversion back to political faction jockeying akin to the high empire, as the various factions can now be physically present to influence the emperors power. This doubles (triples? quadruples?) after the Arab conquest, because now you don't even have rival Mediterranean cities jockeying for power, like the problems 6th century Byzantine emperors had with Alexandrian bishops. Now in Middle Byzantium, there is ONLY The City, and with it, the inability for the emperor to escape these factions who might in the past have been side stepped by simply moving away from them. Thus forcing him to negotiate for power as only a member (though certainly one of the most powerful ones) of the political community.