Analogy: I describe a real-life situation of a police officer shooting a suspect. I then ask people what they think the race of the police officer and suspect are. Because I am referring to a specific real-life case, the question has a single, factual, answer, and people's answer is either correct or not.

Yet I can manipulate the question to show liberal bias or conservative bias, my choice, simply by which case I choose to ask the question about.

The best way to ask that question to legitimately detect bias would be to choose a typical case, and to assume that people who haven't heard of the specific case would answer depending on the facts about typical cases.

And in this situation, a typical case would be an index of X that accurately measures X. Choosing an index that doesn't accurately measure X would skew the ability to use that question to detect bias, since I expect that unbiased people who haven't heard of the index in question would answer based on an accurate measure of X.