There are many many halachic Jews who think they are gentiles. O’Connor’s mother converted from Judaism to Catholicism (apparently for marriage) and never told her son, the Cardinal. So he is from a recent family that left Judaism and yet he never knew he is Jewish. And he became a top figure in the Church, second only to the Pope (as all Cardinals are.)
How much more so the Jews who left Judaism throughout the last 2000 years, whether on their own (as 50% of Sephardic Jewry did in 1492 rather than leave Spain) or did so under force or were kidnapped as children and raised non-Jewish. All their decendents through the maternal line, for thousands of years, are Jewish even though they do not know it.
There very well may be, if not actually likely, many more halachic Jews who have no idea they are Jewish than there are Jew who know they are Jewish. In fact, there are likely hundreds of millions such halachic Jews.
On the opposite token there are many people who think they are Jewish while in reality they are gentiles. This is because of non-Orthodox conversions as well as all the descendents of such conversions. As well as others who somehow got mixed into the Jewish people. Unfortunately sometimes such people become an Orthodox “baal teshuva” and in reality they aren’t even Jewish.