Would a gooseneck kettle (the kind of making pourover coffee) work as a fat separator?

The benefit of a fat separator is that it's clear, so you can see the liquid. With a metal vessel like a kettle, you won't be able to see that before some of the fat pours through the spout. It will be difficult to effectively clean the spout later, especially any grease that enters it in the process. In a pinch I have poured the broth and fat into a container, let the fat rise to the top and skim it first with a spoon and after with a paper towel. That will remove most of the fat effectively if you don't have time to allow the fat to solidify under refrigeration.

I can't say I've tried this, but maybe! If you don't want to use your kettle (and have some time to spare), you could always pour your broth, stock, etc. into a container and allow it to chill long enough for the fat to solidify. Not great in a pinch, but might save your kettle from having any residual fat in it.