I have always found it amusing that people who use this slogan tend to be deeply concerned about bias from the other side but see no problem with such a glib statement.

Because that statement is meant as sarcasm, highlighting the absurdity of certain claims, and not a literal statement.

That's the thing though, I never hear that statement from conservatives or non-liberals - they never seem to utter it, even in a sarcastic or mocking tone.

I only ever hear it from what seem to be sincere liberals/progressives, and often in the context of dismissing claims of liberal bias. In fact, that seems to be exactly how it was used in this very comment chain.