Sounds good in theory. In practice there is one problem with having integrations tests only. The test are generally simple: they pass or they fail. A unit test tests just a small functionality, so when it fails, it's quite easy to find out the problem. When an integration test fails, then we can spend hours debugging the whole stack of layers trying to find out the real problem.

I had this situation once. Every failing integration test ended with hours spent on writing unit tests for all the places used by the test.