Let me say a word about the secondary part of the question:
Any thoughts on why most Japanese people * don't know the origin of the word sake are also appreciated.
A simple answer is because understanding the etymology requires study and research and most Japanese people are not linguists. Ask an average English speaker what the etymological connection is between, say, meal and meat, and I would not expect that you will get any answer beyond a guess. Being fluent in a language does not mean that one knows the etymology.
But there is something more on this. I heard that even when you ask linguists, it is often more difficult to trace the origin of a word in Japanese than in English. This is because the origin of the Japanese language itself is unclear (see Japonic languages and the classification of Japonic). If you think about the etymological study as a kind of detective work, there is often not much clue on which a theory about the origin of a word can be built. On the other hand, there are many languages which have common origins with English, and it is often possible to get some clue by comparing them.
on this. There are a ton of homophones or near-homophones in Japanese which are not necessarily related. Not saying they aren't, but it seems awfully obscure. :)