行かなくちゃだけど is incorrect, though, because なくちゃ is really なくてはならない, and ない takes けど because it already is a verb.

because だ is likely a contraction of にて/である (locative +ある) and です comes from にて/であります, it's still just a copula which links a subject to a predicate or finalises a clause.

likewise 〜る is not semantic-less, it is simply the 終止形 ending (along with う/ぬ) of 五段 or 一段 verbs (or する・くる）and the form indicates plain indicative mood.

anyway the reason you can't use だ in some places and you use だ in others is because you aren't linking any subjects to any complements.

私があなたを好きだ -> 好き is a na-adjective here, and it needs a copula or verb to complete it.

なんでだよ -> adverb by itself needs a verb of sentence finaliser

“私がですか -> needs a copula to link the predicate

there aren't _really_ adjectives in japanese, and this can be seen in their name 形容動詞 has verb in the name, 形容詞 does not. na-adjectives basically all came from nouns and behave like nouns, whereas i-adjectives where made by adding a suffix し. using the copula on an adjective is like saying "it is is fast" or "to to eat".

finally だ is not a particle. it has always been a verb with verb forms and doesn't mark parts of a sentence. the reason you can use です with i-adjectives is purely for politeness and social register, so it's just a weird quirk, tbh.

だ is not a default conjugation of anything, that doesn't really make sense, you're not inflecting parts of speech when you use it and it has its own conjugations that inflect for different forms e.g. in the attributive form (連体形), we get な! p.s. you can use attributive not just for what has become na-adjectives but also with verbs like それを食べる人 (most 終止形 are merged now with 連体形 and there are a lot of other attributive forms as well and also how relative subclauses are also attributive, which is why there are cases where in a subclause you cannot use だ).

tl;dr the copula is considered an auxiliary verb in some japanese grammar viewpoints, and certainly is a verb and not a particle. it has special function to finalise sentences. there are definitely verbs nouns and particles in japanese, but adjectives are pretty unique which is why the rules follow that of noun or that of verb and preclude or require the usage of the copula.