I should clarify: The wart is that programmers can introduce their own unchecked Exceptions. This should not be allowed since, as you say, unchecked Exceptions only make sense for unrecoverable runtime errors.

Checked exceptions should only exist in situations where the caller can definitely recover from them.

False dichotomy. Unchecked exceptions become part of the callee's method signature, i.e. the contract that the caller must fulfill.

Arguably almost nobody does the latter correctly when choosing checked exceptions

Baseless claim. I see many people using Exceptions correctly and elegantly, for error handling and flow control.

and the consensus is

Opinion != consensus.