Attach chat editbox to a frame other than DEFAULT_CHAT_FRAME

[AnrDaemon]

First, a little preamble.
Ever since the days of Lineage2, I'm used to less-than-important-but-still-useful messages to be redirected to a separate, always visible chat frame.
This is how it looks in today's WoW: https://prnt.sc/h9jg76
But here's the problem.
All addons send their output to DEFAULT_CHAT_FRAME naively assuming that it is the frame people watch the most.
There's a simple solution to output data to the desired frame, but when I do something like

if ChatFrame6Tab then DEFAULT_CHAT_FRAME = ChatFrame6 end

the editbox is also become attached to that frame.
Is there an easy way to move editbox to a more appropriate place?