Thanks for the additional information—Rexonola is definitely an off-brand about which I know basically only what you’ve shared with us. From your description, though, it seems they were fairly successful in Australia, and according to Google, they were still making phonographs after Edison gave it up, so they must have been doing something right.

They’re pretty much limited to Australia, though, right? Or have Rexonola phonographs popped up elsewhere?
— MordEth