I agree with you. I wrote an essay once about the need to be sentient about what characteristics of biological evolution one adopts in genetic programming, arguing that sex is arguably just a serendipitous side-effect or at least a sub-optimal solution to the problem of effectively mixing gene material. Much points to this, such as the short-sightedness principle of evolution, and that group or species selection argument is generally hard to make.
However, I am still reluctant to accept your distinction between natural selection and sexual selection. The article you link to even calls sexual selection "a mode of natural selection". The giraffe neck does not have fitness value in the sense of making it a better food processor, quite the opposite. But it does have fitness value in the sense of prospering those very genes, indeed otherwise the giraffe would not exist. Such a definition of fitness does tend to be tautological, but the alternative is no better, because it is inevitably based on a measure of survivability that in the end is necessarily arbitrary.
However, I am still reluctant to accept your distinction between natural selection and sexual selection. The article you link to even calls sexual selection "a mode of natural selection". The giraffe neck does not have fitness value in the sense of making it a better food processor, quite the opposite. But it does have fitness value in the sense of prospering those very genes, indeed otherwise the giraffe would not exist. Such a definition of fitness does tend to be tautological, but the alternative is no better, because it is inevitably based on a measure of survivability that in the end is necessarily arbitrary.