MM literally translates to 2000 in Roman numerals and Roman numerals is always the reason provided for why MM is used for millions. I don’t see where mille or it’s history of steps gets into the picture at all, unless it’s the real reason for MM being million and everyone’s explanation is just wrong (they effectively don’t know why they’re using MM).

It still kind of reasons that if the idea for MM is to be more precise/clear than M, they’ve not done a very good job picking a clearer abbreviation.