n0mesorah,
“What if he refuses to tell us what he believes?”
Well then, he just might not be a good candidate for the job.
“Is there supposed to be some kind of inquisition to make sure people are of the right mind on various critical matters?”
This is an appeal to an extreme. Did I say everyone needs interrogation? And inquisition is a loaded term. We are human beings; we cannot read someone else’s mind. If a Jew is making obvious efforts to follow the mitzvos (living in a frum community, observing Shabbos, YT, kashrus, dressing Jewishly, etc.), then he is assumed to be appropriately suited for basic communal functions. In some cases, we investigate more (or should investigate more), but at the end of the day we have to rely on the answers given. Have some people taken advantage of this (e.g., missionaries posing as frum Jews)? Yes.
“(Deep breath,) don’t ask me to get involved in who you choose as a rabbi.”
Ok, I won’t. But do you disagree with the need for any investigation into the beliefs or character of a man being considered to lead a community? Should we not care about whether the man who paskens Hashem’s law to us actually believes that Hashem exists and gave us that law?