Is "sentence" a useful and/or clearly-defined term in linguistics

Sentence is not a very useful construct for most things you'd want to study about language. It is only an artifact of a particular kind of written language and it is really hard to provide any definition more specific than 'it starts with a capital letter and ends with a period'. Even then, the beginnings and ends of 'sentences' are fairly arbitrary.

The only analytical concept I can think of where sentence is directly useful is measures of readability which look at the number of words per sentence. This can give a fairly reliable indication of the difficulty of text (providing they are long enough). Sentences are also used when dealing with concepts like subordination and anaphora accessibility but they're not really necessary for that.

The two concepts to look at as much more fundamental are:

1. Clauses defined broadly as subject and predicate. They can be found in any language and are much more easily and reliably identified. You can usefully study relationships between clauses in speech without recourse to the notion of a sentence and arrive at very similar conclusions. Perhaps, here the term sentence could be reserved for groups of clauses that are linked through hypotaxis (subordination) but even here, these relationships occur in many non-sentence-like contexts. Like: A: It's going to be a great party. B. Yeah! If Johnny decides to show up.

2. Utterances or texts which are units of expression that can be studied with respect to communicative intent without regard to length or internal structure. Thus, I can treat this entire response as a single utterance/text just as much as the word 'Boo' shouted into somebody's face constitutes an utterance. Looking at the types of utterances, their internal structure and relationships (such as co-reference, turn-taking, repair, etc.) can reveal a lot of useful things about language as a whole.

Ultimately, sentence is just a theoretical leftover from earlier times of theorizing about language based on written sources alone.

However, that does not mean that in many practical contexts such as language teaching, composition, etc. sentence should be avoided as a way to communicate about language. It's a very useful, generally understood shortcut that is probably not even out of place in a textbook on syntax. But in thinking about how language works across many languages and usage contexts, it is of only marginal interest. It just happens that most of the language data available to us is in the form that makes sentences seem like an inherent organizing principle of language.