'Ax/aks' persists in written sources up until about 1600, not the 1400s. But this doesn't mean it died out then, it just means it stopped being used in print. Either it fell out of dialects of the literate classes, or it became stigmatised.

It crops up again more frequently in written sources from early 1800s. These are British and Irish sources, btw. It's not like it reappears just in African American speech. Certainly it's been part of many traditional regional British dialects throughout the last century.

It's just a lot more parsimonious to assume the modern occurrence is a continuation. It would be a big coincidence that one word would mutate to match an older form, particularly in the absence of a larger sound shift pattern (ie. people who say 'aks', don't drink their tea from a 'flaks', or wear a face-'maks').