Is there a rule governing the pronunciation of 'sch-' at the beginning of words in English? Words of German or Yiddish origin are pronounced sh (as in should)but not all words of Latin origin are pronounced sk. And what about the word schedule? Although I am a British English speaker I pronounce it 'skedule' (after Latin schedula) and I would dearly love to be able to prove all those who say 'shedule' wrong.
I know of no rule for choosing between [sk] and [sh] beyond the distinction you outline in your question, with Germanic and Yiddish words with [sh] and Latin and Greek words usually with [sk].
In Old French some Latin words with <sch> and [sk] simplified to [s] with concomitant changes in spelling. Our schedule is a case in point: It comes from Old French cedule, itself from Late Latin schedula. The Latin would have had [sk], which simplified in French to [s] spelled <c>, a pronunciation and spelling that persisted in English for centuries: cedule, sedule, etc.--and remains in Modern French with initial <c> and [s]. In the 16th century the <schedule> spelling appears in English, one of several classical respellings of the tiime, though the pronunciation with [s] persisted well into the 18th century.
I don't think it's possible to prove the [sh] people wrong in any real sense of the word. Both the American [sk] and the British [sh] are spelling pronunciations, though [sk] does have historical priority. Apparently the [sh] pronunciation is by analogy with other French <ch> words with [sh]: chef, mustache, etc., words in which earlier [ch]--that is, [tsh]--simplified to [sh]. One could argue that the British pronunciation of schedule with initial [sh] is a case of stopping too soon in working one's way back through etymological time. There's no particular reason to stop at the French, and working back to the Latin, and even further to the Greek, would lead to initial [sk], as in American English, and in words like scheme, school, scholar, etc. One could also argue that a more legitimate variation would be between [sk] and [s], as with schism, which would get us back to schedule with initial [s], contrasting with American English's [sk]. Whether that makes the [sh] people wrong is hard to say.