To your main point, I wouldn't say exactly that the component stores the state. But, rather that every component provides an initial value, possible events, and a default event handler which is a function from value to value. In effect, this is partially 'storing local state', but the above pieces can be composed to create a container component.

Note that there is no option really - the app wont be reimplementing how a key is handled in a text box. But composability means that the same principle should hold not just for OS/browser components but also for higher level components (A custom map or a tree-view where there are restrictions on types and number of nodes - these should also have default handling and delegation to upper levels.)

The global store choice makes it harder to have component libraries. But, the composable alternative has its problems too - redundancy and communication which skips layers (which requires 'context' in React).