When I've heard people making similar claims, what I've usually found is they're testing "glue" code: controllers, routers, etc. Personally I find this a near total waste of time: it's hard to write the tests, almost never actually catches a bug, and failures in this code are totally obvious during a smoke test - automated or not.

I write a lot of "application" code (cli, service and back-end) and a lot of tests. Parsing, calculations, file generation, regex .. that catches lots of bugs.

The value comes from keeping the complex code separate from the glue, and of course testing it. And you can easily test dozens of cases, which is usually not true of integration tests due to complexity and run time.