I agree with what user6726 has written:
What do you mean by "useful" -- to whom? For what? What constitutes being "clearly-defined" -- do you mean "easy to detect where everybody agrees that the thing is an X"? Substitute any other linguistic term, and you'll get the same range of "yes" / "no" opinions, just from different people.
And my answer is based on your clarification:
I'm supposing the "raison d'etre" for linguistics is to identify, describe, categorise the various features used in language, and the principles/rules governing how they're used. I realise that there will be people concentrating on different things within the entire field, but I suppose I'm asking how relevant "sentence" is to the average linguist. Or maybe I mean, How important is it in the context of "Linguistics 101"?
To the last question, I'd say it's eminently important. When I searched on google for "linguistics textbook pdf", I came across Introductory Linguistics -- Bruce P. Hayes, The Study of Language -- George Yule, Introduction
to English Linguistics --László Varga, and so on. In each of these, the concept of "sentence" was used without much of an explanation, relying on its audience to already have a clear conception of it. The last book I cited distinguishes between sentence and utterance as follows:
Sentences have to be distinguished from utterances. A sentence is any string of words produced by the sentence-forming rules of a language, these rules are stored in native speakers’ competence. (By competence we mean the native speaker’s intuitive knowledge of language, see Unit 2 above.) So sentences are constructs of competence, they are ideal, abstract entities. For instance, Peter smokes cheap cigars is an English sentence because it has the structure of an English sentence.
By contrast, an utterance is typically the physical realisation of a sentence in a real situation of language use, i.e. in performance. (Performance is the actual use of competence and it involves individual and situational factors, see Unit 2.) Since utterances belong to performance, in spontaneous speech they often contain imperfections, such as hesitations, false starts, lack of concord, etc., especially if the speaker is tired or excited or embarrassed.
My only point of contention with the phrasing above is that I would rather have said "an utterance is the spoken realisation of a sentence". Afterall, a sentence is physically realized whether it is spoken, written, or gesticulated (as in ASL), and none of these are any less real situations of language use.