It looks like while ChatFrame1EditBox reports a "DIALOG" frameStrata, it's rendering as if it was on the same frameStrata as ChatFrame1 (its parent), or "BACKGROUND". MainMenuBar is "MEDIUM".
Code:
/run ChatFrame1:SetFrameStrata("DIALOG")
seems to move the editbox (and chat window) above the bars.
This looks like a bug, but maybe needs more testing to see if it's consistent behavior beyond these frames.