I recommend this paper by Ridouane which argues that in Tashlhiyt, stops can be syllable peaks (are syllabic), for example [tb̩.dg̩] 'it is wet'. This question has been addressed in the literature: the advantage of this paper is that it is written with full knowledge of that preceding literature. I would also recommend Bagemihl's paper "Syllable Structure in Bella Coola", because it engages a number of alternative theories of syllabification for the Nuxalk facts: obstruents are not syllabic in Nuxalk.
Reading through especially Bagemihl's paper, it should be clear that good argumentation establishing how consonants in a big sequence are sylabified can be made based on phonological patterns, but it is very difficult to make that argument, and it may not be possible. Since "psst" isn't a word of English (it's a sound conventionally made by English speakers), similar to "pfffft", "d'oh!" and [txŇʔn̥] (the nasally-released snort), there's nothing in the grammar that you can call on to decide if these things are syllables or what.
In light of Ridouane's evidence, I would say that the reason why [t] is not a syllable peak in English is that that's the way the rules of syllabification in English work. Some languages allow sonorants to be syllable peaks, some don't; some only allow liquids to be syllable peaks, others are less restrictive. There are functional explanations that say why usually, only vowels are syllable peaks, but the specific patterns of individual languages have to be specified in the grammar of the language.