>By far the most imprecise language I know, and I suspect this is by design (politeness), is Japanese. It basically has no subject and most of what we'd call grammar is derived from context.
First, the obligatory, any claim that a particular cultural element causes a particular language feature needs tremendous evidence; languages evolve on their own.
Second, Japanese has very explicit subjects, they are what comes before the は or が particle. By convention, stating the subject in a sentence is optional and generally omitted if it could be understood from context. You could argue that this is an imprecision, but if you need the precision for something, then just state the subject. There is only a 1 syllable overhead for doing so (beyond the syllables in the subject itself). From a technical (read linguistic) standpoint you could argue that what is marked by は and が is not really "subjects", but something else ("topics"), which is technically correct[0]. One could also argue that Japanese does not have adjectives, which is also technically correct, but both of these say more about how the field of linguistics chooses to model and define terms than about Japanese itself.
Third, Japanese has a very well defined grammar. Unlike English, it is largely not based on word order, but by particles that follow phrases to indicate what part of speech said phrase is (and yes, in my experience, this is very useful for poetry). Admittedly, colloquial Japanese sometimes omits these particles, but when they do so the sentences are arranged in very standard forms turning it into a word order based grammer.
[0] Mostly. Sometimes は and が really do mark the subject.