Conversion has always been a part of Judaism, except at certain times when leaders prohibited it, such as Ezra and Nehemiah.

Jesus was descended from converts Ruth and Rehab.  I read that only three women are specifically named in Jesus' ancestry,
and two of them are the converts Ruth and Rehab.

It would be overwhelmingly hard, although probably not impossible, to find a "pure" Jew today, one who is descended from
Judah, Benjamin, and Levi....the others being descended from the other tribes and mostly living in the northern reaches of
Israel in Bible times.  Remember that ALL the tribes left Egypt with Moses and all entered the Promised Land eventually,
although many individuals of each tribe died on the way.  And with these people came many converts, people who chose to
leave Egypt with the Jews.  

Now you certainly have something real when you say that many "Jews" today are not spiritually Jews.  Those who follow the
Illuminati cabal are not spiritually true Jews.  It is amazing that they call themselves Jews at all, since they are so intermarried
with wealthy gentile families.  At least this is true, I read, of their "leaders".  

When I was a young girl, I knew many Jews because they ware in my high school and university, and because for some reason
the young men were attracted to me.  One of these young men whom I knew best was descended from an offshoot of this original
Illuminati family.  He was not "good news".  Praise God for escape from that trap.

My second husband (first died) was half Jewish by descent, including descent from Levi, but he was never raised a Jew.  He
attended a Christian church in his small Oregon town, and was  Methodist song leader at University.  He always said I knew much
more about Judaism than he did.  His daughter from his first marriage, however, married a Jew and went through conversion to
Judaism.  Her children are Jews, especially her last child who is strongly anti-Christian because she thinks that we are "Jew-haters."
Her first child married a well-known Jewish journalist (known in the Jewish community) and has a Jewish child.  Jews I have met were
always puzzled when I told them I was the wife of "X's grandfather".  They thought it very strange that I, a non-Jew, could be the
wife of X, who must be a true Jew to be the grandfather of X.  Many of these Jews who thought this were blond people who were
obviously not descended from people of the traditional Jewish homeland in the Middle East.

These people are considered to be true Jews by the Reform Jewish congregation.  They would have to be re-converted if they
became Orthodox, going through certain rituals such as the Mikvah, (bath or baptism).  

As for the converts from Khazaria, usually referred to as Ashkenazi Jews, they too are all mixed up with the Sephardic Jews.
When I went to high school, many many of my classmates were descended from Sephardic Jews from Rhodes, but one of these
Sephardim had the surname "Ashkenazi".  This probably means that he was descended from an Ashkenazi Jew who married
a Sephardic Jew.

Many of the persecuted Jews of Spain, in the middle ages, were also converts, surprisingly, but history teaches strange lessons.
Some of these converted jews, who were of course persecuted by the Christians, migrated to Mexico and Scotland as places of
refuge.  Scotland became a truly "open" and welcoming home to Jews in the Middle Ages, (and to Moslems and others), who were
usually forced to conceal their identities "pro forma", but were actually known to be "secret Jews".  Some of my ancestors from
Scotland have names which indicate that they were of "secret Jewish" descent, and it is even historically known positively that
these ancestors migrated to Scotland in the middle ages, from England, before that from France, and then their trail gets lost in
various European nations.  Some Scottish "secret Jews" migrated to the southeast part of the United States, and formed Presbyterian
churches, but were also mindful of their Jewish roots.  Those who married Indians there came to be called Melungeons; two
famous men of part Melungeon descent were Abraham Lincoln and the actor Tom Hanks.

The bottom line to me is that "true Jews" are those who follow HaShem and try to do His will, however difficult or impossible that
may be.  Often they are hostile to Christians because they believe it is impure to worship a man, such as Jesus.  They do not
believe Jesus is God.  However, they do follow the G-d of Abraham, Isaac, and Jacob, and for this reason are called Jews.
And so many of them have gentile ancestors that it may be almost impossible to find a "true Jew" in the racially pure sense.
Probably there are a few in Brooklyn in a Hasidic congregation.  Probably even a few who might have a "genealogy" to prove it.