This reflects my own view of testing. Namely, that the most important thing with a test is to detect and prevent mistakes.
Most mistakes are not made on the unit-test level, but rather when plumbing things together into a whole. It is much, much more common to break the entire site through a REST API changing, and the client not understanding the backend any longer. Or that an integration with a third-party fails after either party updates their end, than it is for an error to occur in a single component.