Automatically generating test cases for software has been an active research topic for many years. While current tools can generate powerful regression or crash-reproducing test cases, these are often kept separately from the maintained test suite. In this paper, we leverage the developer's familiarity with test cases amplified from existing, manually written developer tests. Starting from issues reported by developers in previous studies, we investigate what aspects are important to design a developer-centric test amplification approach, that provides test cases that are taken over by developers into their test suite. We conduct 16 semi-structured interviews with software developers supported by our prototypical designs of a developer-centric test amplification approach and a corresponding test exploration tool. We extend the test amplification tool DSpot, generating test cases that are easier to understand. Our IntelliJ plugin TestCube empowers developers to explore amplified test cases from their familiar environment. From our interviews, we gather 52 observations that we summarize into 23 result categories and give two key recommendations on how future tool designers can make their tools better suited for developer-centric test amplification.