In my experience, the issue is always the use of "flush" when "flesh" is meant, never the reverse.
"Fleshing out" is adding details, specifics, or clarity — the meat on the idea skeleton. "Flushing out," on the other hand, means more along the lines of forcing an idea out of hiding (or flushing bad ones down the proverbial toilet).
The phrase "flushing out a character" immediately brings to mind a wild-haired Baba Yaga, waving her broom and yelling, “Come out, protagonist! I know you're in there with your half-baked backstory!”