"...syllables which were distinguished by the Linear A language, but not by Greek - such as aspirated/unaspirated P." Given that aspirated and unaspirated voiceless stops were almost certainly distinguished in spoken Greek of the time (as they were in proto-IndoEuropean and in later classical Greek), why would the Greeks not have carried over such a distinction if it existed in the Linear A language? It seems much more likely that the distinction did not exist in the Linear A language or script, and that's why it didn't show up in Linear B.
True, I did not think of the pi/phi distinction when writing this. There are other quirks exclusive to Linear B that don't occur in Linear A though, such as three different ways to write the phoneme "A" with no clear pattern on when is which one used.
With regard to the origin of the script, Linear A documents have been dated to earlier times than Linear B. And then, there is also an even earlier hieroglyphic script, but its relation to Linear A has not been established.
