My original question, though, is not answered yet, namely why not to
detect that we are about to pattern match on a GADT constructor and
allow the programmer to capture the *refined* type with her type
annotation. Sure this would necessitate a change to the type checker,
but would also increase the expressive power a bit.

Is there some fundamental problem with this? Or simply nobody wanted
to do this yet? Would it be hard to implement type checking *after*
refinement on GADT(-like) patterns?