The problem that you end up with past a certain scale in a component-based app without a state management framework (third party or your own) is that you're performing business logic inside view components. By moving your business logic into your state management framework and using something like redux-saga that is designed specifically for managing complex asynchronous workflows, you can extract your application code from your views and make your views pure functions of the state tree.
When I hear someone complaining about redux using global state, I like to ask them what makes global state bad. The answer is usually that it allows any part of the application to change a value, which makes debugging difficult. This doesn't apply to redux and similar frameworks because every slice of the state can only be managed by a single, pure function, which makes it incredibly easy to debug. Additionally, redux uses a copy on write mechanism, so your state tree isn't actually being mutated, which allows you to use equality comparison instead of a deep compare to see what state has changed. That, combined with the fact that all actions go through a central dispatcher that is easily audited means unit testing your state transitions is almost laughably trivial. And since you are pulling all of your business logic out of your UI components, you can avoid the side-effect hell of unit testing your UI.
Having said that, I won't use react or redux for any website that doesn't have a heavily interactive UI with complex state that must be shared across many pages. Managing application state in your front end is an enormous burden and you'd be crazy to take it on without a need for it.