Maybe I'm just trying to apply western principles to Japanese grammar and my question doesn't even make sense. I'm not sure.
Let's explore that possibility. I agree that the grammatical terms used to describe English (along with most European languages, etc.) don't fit well to describe Japanese.
The terms "noun" and "verb" are fairly universal, but even "noun" has complications (there are many grammatical properties you might conceivably expect a "noun" to have, and multiple overlapping classes of words in Japanese that have some but not all of those properties). By the time you get to "adjective" there is already no clear agreement.
What we call "i-adjectives", i.e. what are natively called [形]{けい}[容]{よう}[詞]{し} (literally "description words"), function in most respects the same way as verbs ([動]{どう}[詞]{し} "motion words", even though this includes completely stative and intransitive verbs). In particular, they are both predicates, and they both can be used attributively.
Here by "predicate" I mean that by combining with just a grammatical subject (and we can even omit this contextually in Japanese), we get a grammatically complete (even if semantically vague) utterance. In English, we cannot say "the cat cute"; we require a copula in order to turn the adjective "cute" into a proper completion for the sentence. But in Japanese, 猫が可愛い is fine. The particle が is not a copula, but only a case marker. We do not require です; and if we add it then it is functioning as a politeness marker, not as a copula, which is why we can't substitute だ. The copular function - the part represented by "is" in English (although this has many other uses in English) - is already contained within 可愛い.
On the flip side, English does not allow us to use verbs to describe nouns directly (attributively): we cannot say "a walks person", but instead "a person who walks" - the verb needs to get pushed into an explicitly marked relative clause using a relativizing pronoun. But in Japanese, 歩く人 is fine, in the same way that 可愛い猫 is fine.
Mainly where these predicates differ is that 動詞 generally accept a full set of case-marked participants in the "action" - が-marked grammatical subject, を-marked grammatical direct object, as well as に- and で- marked participants for which analogues are not as clear-cut. When used attributively, this implies that something that would be a complex sentence by itself can then be used to describe a noun by just dropping the noun after it (meaning something like "the (noun) for/in which (complex description of event or premise) occurs/is the case"). 形容詞 on the other hand, as far as I can recall seeing, only accept a が-marked subject; and when used attributively, such a subject is generally part of a fixed expression (e.g. 背が高い).
Is ある->ない just a special case?
There are many ways to analyze this. My view is: yes, but it's an irregularity of the conjugation of ある - not a separate kind of ない. In Japanese, the negations of 動詞 simply are 形容詞 that are formed using ない.
Whether the [未]{み}[然]{ぜん}[形]{けい} ("a-stem" for godan verbs) to which ない is attached counts as a "word", let alone a "separate" word from the ない itself, is a matter of definitions, and not a fruitful debate; but ない by itself does negate ある, and it is the same thing as ない attached to a 動詞 in 未然形 form. It just happens that ある is irregular, such that its "a-stem" is null when attached to ない (although the volitional, ある -> あら+い -> あろう (via ウ音便), seems to be regular). Alternately, we could say that ある is "defective", and the verb ない is substituted as its negative form, in the same way that できる is substituted as the potential of する.
Describing ない in English as an "auxiliary verb" doesn't make sense if we're going to use the word "adjective" at all. Verbs conjugated with ない end with... い, i.e., not a う-row kana; and they don't accept a を-marked direct object etc.
The remaining question is why the conjugation works this way. To my understanding, the mental model is that only the underlying "action" itself is being negated, not the entire proposition. In English, it's natural to say something like "I didn't tell the story", and "the story" is still a direct object. But in the Japanese mindset, if you didn't speak, then the unsaid-thing no longer merits the grammatical privilege of being marked with を. So either the former direct object has to become a topic, or else the entire proposition can be nominalized and its existence (although this doesn't feel like quite the right word in English) then denied.
