If I trust my Latin dictionary, psallo means playing on the zither and derives directly from Greek ψάλλω. The letter ψ also remains in numerous words of Greek origin like psychology. There is little reason to assume, that the p was ever silent in Roman empire or Greece.
Wikipedia states, that church Latin moved its pronounciation towards Italian (most noticable in combinations like ce, ci, where the c is currently assumed to have been pronounced as k by native Romans, and ge, gi). Italian with its simple pronounciation has practically no silent letters (aside from "h" in combinations like ghe). Psicologia (Italian for psychology) is pronounced with ps.
This leaves two possible (somewhat questionable) reasons:
- English pronounciation drops the p at the start of Greek words (on the other hand English has not the best track record of preserving original pronounciation). See related English language forum answer.
- It is awkward to sing
For sure it makes no difference, whether ps is at the beginning of a word (or sentence) or somewhere in the middle.