A bit late to this very interesting question which had also troubled me for years to find the answer. My answer for this question is that it's simply the way those sound are pronounced in English! That is English plosives are unaspirated when they immediately follow /s/ because that's how English speakers do that. So that makes /p/, /t/ and /k/ unaspirated in English /sp/, /st/, /sk/ consonant clusters.
Of course, you can pronounce /sp/, /st/, /sk/ with aspiration, there's no rule telling you that you can't do that and in fact everyone can certainly do that without difficulty. But that's not English /sp/, /st/, /sk/.
In a similar vein, you can also use the same argument to explain why /p/, /t/, /k/ are aspirated when they're at the beginning of a stress syllable.