In classical Greek, the only consonant digraphs that have corresponding letters in the alphabet are ks(Ξ), ps(Ψ), and zd(Ζ, though this one's debated) -- why aren't other consonant digraphs (e.g. ts, kp, kth, sz) identified with distinctive letters? I thought it might had something to do with their frequency of use, but I am not certain. It'd be very helpful if there are some sort of frequency statistics of all consonant digraphs available.
My feeling is that the "doubleness" is a bit of an illusion, conflating letters which seem "double" for different reasons.
Ζ is a perfectly respectable single letter, a cognate of the Hebrew zayin. If the Greeks happen to find themselves dissimilating it in pronunciation, that is a secondary phenomenon.
Ξ is, similarly, a single letter, a cognate of samech, which is one of those Daffy Duck-style sibilant consonants which Semitic languages like and Greek doesn't. That a "splashy s" becomes a "ks" is not unparalleled. The letter "x", for instance, in parts of Iberia and also in Mexico (even in the word "Mexico" itself), is used for various kinds of "sh" sound. Again, this is a single letter for which Greek happened to find a double-consonant pronunciation.
The letters φ and χ (which you didn't mention) are clearly late additions because they come after the end of the alphabet. They had to come into existence because (a) aspirated π and aspirated κ mean something different from their unaspirated versions and so need to be distinguished in writing and (b) after quite a lot of vacillation Η turned out to be more useful as "long Ε" rather than as "aspirate". (That, again, is not unheard-of. In some accents in French the final vowel in oui has a definite "h" sound.)
Of the digraphs you mention, κπ, κθ, κφ are not true consonants, just the collision of a κ at the end of one syllable with another letter at the start of the next. But there are better examples. κτ, πτ and σθ, for instance, are genuine consonant clusters that can occur at the beginnings of words. The first two of these, at least, have a chance of being genuine "consonants-in-themselves", hybrids of their components and not just one sound after the other. Compare the "gb" in "Igbo" which is not two sounds but a single sound articulated in two places at once. If one were evolving a "rational Greek alphabet" those would be good choices for being made a single letter; but it didn't happen.
All this doesn't answer your question so much as sharpen it. Why, out of all the conceivable consonant clusters, did it become desirable to write ΦΣ as Ψ and to make no other change? I say Φ rather than Π because I think that inscriptions used to use ΦΣ rather than ΠΣ. The resemblance between Φ and Ψ also makes me think that this new letter followed a general rule for adding letters to an alphabet, which is not to add new and unprecedented shapes.
Was it, perhaps, something to do with the special character of the S sound, making it, like the H sound, feel more like a modifier of its preceding consonant rather than a consonant in its own right?