How about this: of course fitness is a phenotype because you can measure it (with error) like any other phenotype and it is the product of an interaction of genes and environment. But it's special, so it usually doesn't make sense to treat it like any old phenotype...

...except in certain modeling situations (Fisher's fundamental theorem, etc.). It's only special because it is the ultimate predictor of evolutionary change, not for any of the other reasons people have listed.

Fitness is the product of other phenotypes, but so is like every other phenotype. You can write genes > phenotype > fitness or genes > physiology > morphology > performance > fitness or whatever, and none of those things are inherently special because of where you wrote them.

We can't break organisms or the world in general down into neat tidy packages that don't overlap. Biology is complex!

Anyway, there are lots of valid ways of seeing things depending on what you're trying to understand or model at a given time. But it's fun to argue about stuff on Twitter. Thanks to @PetrovADmitri for stimulating some discussion!