website: Improve reliability of e2e tests

[othrayte]

The main change is to useGenericTaskAPI() to have clean cases when the task is set and when it is awaiting progress. I also changed the frontend to pass through the 503 code we get when tasks are not available, but then decided that "no tasks available" is better than just freezing up on other errors, so all errors are currently handled the same.

Some other type consistency changes were also make to cleanup the task types and also some of the prop names.

[othrayte]

Oh look, the E2E tests passed. 😆

[fozziethebeat]

I'll leave the review to @AbdBarho