Most people in the historical sciences in Western Europe and its former colonies do not believe that it is possible to do what Turchin believes he can do, because we don't know enough about enough of the past and because long-term predictions of a mathematically chaotic system are impossible. Cliodynamics fits more into a Russian tradition of grand mathematical theories of history than into the historical sciences in other parts of the world.

Historians and archaeologists have engaged with some of his books and his SESHAT project, often criticizing the underlying data and the structure of his arguments. This blog post has links to some of the journal articles and blog posts by both sides, you can find others like this one by Duncan Keenan-Jones and Owen Hebblewhite in his own journal Cliodynamics. But in general, the kind of thing the Cliodynamics believers hope to do is not the kind of thing historians or most kinds of archaeologists are interested in; historians are trained to focus on the specific and what makes each case unique, not on finding general laws. That does not mean that the "general law" approach is invalid but its not where most historians focus their research time.

There are a few historians and archaeologists in the United States like Michael E. Smith, Ian Morris, and Walter Scheidel who work on quantitative ancient history, but their claims tend to be more modest and built around things we can actually see in the historical and archaeological record (shipwrecks and coin hoards, not GDP or population). Works in this tradition appear from major academic presses and researchers from other traditions engage with them, you could call it a respected minority view. Turchin and the journalists don't always talk about this tradition as much as they could, because he likes presenting himself as a heroic outsider and a leader of a revolution.

Edit: and yes, Scheidel and Turchin have co-authored publications like "Coin hoards speak of population declines in Ancient Rome," PNAS October 13, 2009 https://doi.org/10.1073/pnas.0904576106 but as far as I know Scheidel does not use his methods to predict future events to an accuracy of a year or make broad statements about 10,000 years of world history. The historians and archaeologists who are interested in quantitative methods tend to have less ambitious goals like "use house sizes as a model of income inequality in Pre-Columbian Mesoamerica" or "compare the economies of the Han empire and the Julio-Claudian empire" or "test whether colonialism in India was a major contributor to the First Industrial Revolution in Britain."