Fair, but not really. I used a wrong word. While I agree that your response is correct in pretty much every respect, it highlights the consequence of rudimentary practices in a de-centralized field.

To the first point, "the term hybrid is used for this very reason, as juxtaposed with species." Obviously, I wasn't clear enough in my meaning that you felt the need to point that out, so I'll just own that mistake.

As to the rest, I'm sure you know "species" denotes "a population capable of producing viable offspring." So why would we ever use a different word? Outdated understandings lead to outdated naming conventions which persist after the science has surpassed the action. A hybrid, by definition in the context of biological evolution, is "offspring produced by more than one species." That's an explicit indication regarding the presence of distinct populations or a statement about the capacity to produce viable offspring. One of these criteria is affected, otherwise they should just be called the same species, sub-species or maybe even same ring-species from the very get-go.

Like taxonomy as you mentioned above, asserting primacy to the factor of a population is more on the arbitrary side of things. It's a blurred line in many respects. Viability, however, is not.

Most people tend to think of species as a label. A is this, B is that, etc etc. Consequently, they gloss right over the correlating feature that A is NOT B, B is NOT C, and so on. The underlying question is "Are A and B the same species, or of different species?" Well, it's science, and we can test that, or at least keep our eyes peeled for a test performed by nature. Are they able to diffuse alleles into a receiving population without impediment? That's a closed-ended question. While the answers to that tend to be found somewhere along a spectrum, it's a tighter constraint on subjective opinion, and is consequently the preferred approach. Veering the other way by calling mixed-plant offspring "hybrids" is, for lack of a better description, kind of missing the point in so many ways. That's a practice rooted in history, but not so much in the core of science, which is testability/falsifiability. Since the discovery of DNA and the commercial availability of DNA tests or other advances in tech/science, there's hardly an impetus to use that more-arbitrary decision making process as a place-holding crutch until such a time as a verifiable answer becomes possible anymore. We can police that up, we just typically don't.

We tend not to go back and re-label things accordingly chiefly because it conflicts with naming rights, plus the fact that it's a rather large undertaking that would require updating an entire field. That's pretty much it.

With respect to the viability of f2 generation hybrids, eh... re-classifying the parent generations in the manner described above erases the issue you present, which is essentially what a debate surrounding the relationship between humans and Neanderthal is all about. That all boils down to motivation or a sense of importance to "get it right" rather than a question of if hybrids are viable or not, or if our definitions need to change at all.

So yeah, "hybrid" absolutely does imply that the offspring are not viable. We just incur some leftovers because the history of taxonomy is loaded with examples that represented the best effort possible at the time, which in turn leads to a lot of people still doing it now.