That’s more a consequence of hooks & context in React specifically , rather than the component model. There are other frameworks where it’s much easier to reason about updates.
React encourages you to push state down, into components low in the tree, as seen in this article, which leads to "state and templates being in the same thing". But that is not a property of the component model; you can keep most of your components pure if you'd like, and keep the state and logic at a higher level. At that point there is little distinction between components and "sub views" in a template system. It would just be quite painful to work with.
I do agree with your other point - it's almost impossible to use React as 'just a view library', it ends up permeating everything. I believe the hooks API aggravates both of these problems: the former by making it easier than the alternative patterns, the latter by creating function 'colors' that are contagious.
