Let's work this out. Whall implies that it was known in public schools in mid-nineteenth century. This suggests (both the timing and the location) a transfer via Christy Minstrels or something of that ilk.
I suspect his "Originally it was a song, not a shanty..." is wishful thinking. He thinks "shanty" implies "sea" and "Missouri" implies "not-sea" so.... . The difficulty is that it implies adoption of a shanty at a very late stage??