I looked it up, and apparently it originated in the American south, some believe in Texas. The original meaning of cute was "acute" so it maybe makes more sense that way. The meaning of the word cute started changing from sharp to adorable around the late 1800s. https://www.phrases.org.uk/meanings/as-cute-as-a-bugs-ear.html
I agree with everyone saying they've heard 'dead as a doornail' and 'cute as a button' over 'dead as a Dodo' and 'cute as a bug's ear'