In your response on converts as judges, it seemed to recognize the Jewishness of the father of the convert. This was surprising. I thought a child of a non-Jewish mother lost all Jewish connection with the father and is a Gentile in every respect.

The taking into account of the Jewish father (which you are correct, usually has no ramifications regarding the gentile child, even if he wants to convert) is just one factor, which some explain may apply in this relatively rare case where a convert actually "works his way up" all the way to become a dayan, and helps us just regarding the possible explanation I mentioned that "some suggest that leaders should have the most impeccable lineage possible, in order to command respect". Aside from this relatively minor point in a relatively rare case, you are correct that the father's Judaism is usually irrelevant, e.g. the converted son and Jewish father can even testify together and for and against each other as if they were strangers, something which is totally unacceptable if they are considered "relatives".