I've never understood the "testing little things in isolation" strategy when it precludes a "test everything all working together" strategy.

It doesn't. TDD is largely about how to use unit tests to drive incremental development, but it certainly doesn't preclude any other form of testing being part of the lifecycle (it just doesn't have anything to say about them -- presuming that you have some integration and acceptance test practices, and that those are out of TDD's scope.)