Genetic Jewish Identity

Upon reading Rebecca Alpert's "What is a Jew? The Meaning of Genetic Disease for Jewish Identity", I was surprised to read about the contemporary views of Jewish identity. She mentions the standard Halakhic definition of a Jew is either your biological association (mother being Jewish) or religious transformation(conversion). This is consistent with what we discussed in class previously, about who is defined as a Jew, more specifically we have learned that even with religious conversion, some are not respected as a Jew unless their mother is a Jew.
Therefore, I was surprised to read about the contemporary view that Alpert mentions. "Many contemporary religious Jews also reject Jewish identity based on lineage as they are uncomfortable with secular Jews claiming Jewish roots based either on nationality as is the case in the state of Israel, or on ethnicity as is the case in the United States and other countries"(3). Given that they dismiss the genetic aspect, they are concerned with a religious aspect. Clearly if you are born into a Jewish family you don't have to convert, but your Jewishness is based on your connection to G-d. We have discussed Jewish culture in this class quite a bit, and I for one have always respected any individual claiming themselves a Jew based on their own interpretations of the religion. And while I respect the opinion that being a Jew should be based on a theological connection to G-d, I cannot fully agree. Besides for attending services at a synagogue and having a Minyan, someone can pray or have a connection with G-d on a personal level, without ever speaking aloud in front of anyone else. Therefore, how can someone else judge one's connection to G-d?