Oxford dictionary is not a primary source, and if I'm not mistaken, they subscribe to the descriptive persuasion, i.e. collecting lexemes as they are in use, especially the online version, and others use it as prescriptive source.
All I see is "I always thought it was make do, because that fits better into my primary vocabulary and I don't speak French so I don't know how to pronounce dew, I mean do, err I mean ... that's just like your opinion, man".
I liked the comment noting the uptick (up take?) of make-do after WW2 much better -- which coincided with the invasion of often underedumacated Americans into Europe.
Here's a nice quote
... which shows how careful the Poet was to make due provision for his amendment.
If you had met due dilligence and followed through from the ngram search I linked before, you'd find plenty more sources from before 1660 like that, one with make do from 1660, and plenty american usage after 1945.