As I understand, grammars boil down to the acceptability problem: Is an utterance acceptable to the users of some language X? How to differentiate acceptable from non-acceptable utterances? Those who offer a grammatical analysis of some language X aim to describe the rules that differentiate acceptable from non-acceptable utterances.
Different approaches then differ only in such aspects like the (1) format of the description (2) the definition of acceptability (3) "area" of language covered. For (2) for instance, some define acceptability as merely grammatical, and ignore the issue of semantic well-formedness, while some include the latter in their definition. For (1), some require a description using a language that is easily implementable in NLP, while some settle with natural language descriptions.
For (3), some only deal with acceptable sentences (traditionally "syntax"), while some widen the scope and deal with acceptable words or word forms (what we traditionally call "morphology"), and even acceptable sound combinations ("phonology") or graphic representations ("orthography").
I also am cognisant that some approaches might even be explanatory and predictive rather than descriptive. They aim to explain why certain acceptability rules exist based on non-linguistic factors. And that some might be prescriptive, aiming to promote a certain rule, rather than to describe what rules actually exist.
(1) This is what I understand, but I'd like to know if I am missing something: are there grammatical approaches that do/did not aim to deal with the acceptability problem at all in the first place?
(2) Also, am I missing something even within the acceptability problem approach?
My aim is to have the most inclusive view of all that has been done on grammar - all schools of thought, all languages - as much as possible.