Thinking about this, at the moment, I don't think static typing is a replacement for testing (or vice versa!). Although, apparently, with something like LiquidHaskell, you can get more logic "into" the types and checked by the compiler, but I'm unsure how much you can do when it comes to more complex business logic.
Regarding testing "glue": static typing often gives evidence (but not proof) that code is glued together appropriately [refactoring even small projects without tests in Haskell is a joy: the compiler essentially tells you what to change]. However, it doesn't give evidence that the high level behaviour is what it needs to be. So I think higher level tests are still needed.
I think maybe changing the first question from...
> Am I confident the feature I wrote today works because of the tests and not because I loaded up the application?
to...
> Am I confident the feature I wrote today works because of the tests and type checking, and not because I loaded up the application?
will probably help you to answer the question about how much static types allow you to forgo tests. My instinct is that in most cases, high level tests are still worthwhile.
I'm thinking much the same, though I do think of tests as looking after the dynamic behaviour and types after the static behaviour. Which seems obvious in retrospect, but once you wrap your head around it you can build neat abstractions like lightweight static capabilities: https://github.com/yawaramin/lightweght-static-capabilities
Regarding testing "glue": static typing often gives evidence (but not proof) that code is glued together appropriately [refactoring even small projects without tests in Haskell is a joy: the compiler essentially tells you what to change]. However, it doesn't give evidence that the high level behaviour is what it needs to be. So I think higher level tests are still needed.
I think maybe changing the first question from...
> Am I confident the feature I wrote today works because of the tests and not because I loaded up the application?
to...
> Am I confident the feature I wrote today works because of the tests and type checking, and not because I loaded up the application?
will probably help you to answer the question about how much static types allow you to forgo tests. My instinct is that in most cases, high level tests are still worthwhile.