The number of exceptional errors one can reasonably handle is so small to be insignificant. Aborting the current operation (which could be just one task or the whole application) is the most common result. Abort and log. Unchecked exceptions make that easy. Checked exceptions make that long and tedious and add nothing.

What and where you can handle exceptions has nothing to do with where the exception is thrown. Handlers only exist at key points in the application at the start of operations where you can skip, abort, or retry.