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What makes them Jewish in any sense then? Understand, the Jews of the bible had temples, a priesthood, and offered sacrifices (namely for the atonement of sins). This is covered in Exodus 25-40, Leviticus, and Deuteronomy. In fact it makes up the bulk of the Mosaic covenant.
Since the destruction of the temple in 70AD, jews have abandoned all of this. They no longer keep the Mosaic covenant. So, factually, they are not the jews of the bible. The Talmud was an attempt to reconstruct Jewish identity through Rabbinic dialogue.
As you say, many jews don't follow the talmud, but they also don't follow the torah. So what makes them jewish?
Jewishness has come to be defined ethnically. Judaism is the only religion that claims to be also be a race. And that's not really a good thing.
We have reached the ugly center. Jews believe they are Gods chosen people. If that is not based on any religious pretext, then the claim amounts to ethnic supremicism. The exact same as the nazis.
Does this mean all jews are bad? No. But up until 1965, the Catholic Church openly sought the conversion of jews. After Vatican II, they abandoned that pursuit. Christianity has lost every single battle in the culture wars to Jewish influence since. Abortion. Pornography. Feminism. Multicultuarlism. Sexual liberation. These are but a few examples, but Jewish influence and teaching was at the center of each movement.
What's the solution? In my opinion, a return to doctrine of Sicut Judaeis Non: we are not to harm jews, but neither are we to allow them to harm our culture and society. On good article on the so called Jewish question, written in 1890 and published in the official Vatican encyclical can be found here: http://catholicapologetics.info/apologetics/judaism/jewrope.htm
If you want to gain a complete picture of the historic and theological problems, pick up Dr E Michael Jones' book The Jewish Revolutionary Spirit and it's Impact on World History.