In this short answer, I have broken up the answer into tow parts for clarity.
You ask, "If a human gains a Jewish soul at conversion" - there needs a correction. Non-Jews do not gain anything. If so, did Abraham have a “lesser” soul then Moses? Is the convert a new kind of man? Is G-d tribal, caring less for all the rest?
In fact, the Jew does not possess a special soul. Gentiles share the same soul. His conversion adds nothing new. Ovadia Yosef may have held some disdain for gentiles but Ruth was a gentile who converted and Kings’ David and Solomon, as well as the Messiah, will descend from her. We see from here that gentiles have the same capacity to become the greatest leaders as those who are born to Jewish parents.
Jews cannot convert back to Judaism. Once you are born (or converted) you are always a Jew and will die a Jew. This is why it is imperative to (at first) drive away future converts because once one coverts, you cannot leave the faith.
In Judaism there is a concept about the neshama yesara; that is, an additional soul added for Shabbos in order that we shall channel our energies toward G-d. So even if you break Shabbat; G-d forbid, the soul remains the same but, there is no addition soul unless you keep it. Is that a good incentive?
Simply put, there is no better birthright, only accomplishments. Once a Jew is born (or a gentile converts), he keeps the same soul . Likewise, Once a gentile converts, he keeps the same Torah as a Jew. Thus, all people, are created equal.
 See (Deut. 4:6-8).
 See (Numb. 15:16).
 Talmud Sanhedrin 59a states: “A gentile who studies Torah is akin to a High Priest.”
 Talmud (Shavuos 39a) does not say that Converts have a Jewish soul. It says, “future generations of Jews, and gentiles who will eventually convert, are also part of G-d’s covenant.”
See Maimonides, "Guide for the Perplexed", and Saadia Gaon's "The Book of Beliefs and Opinions" for more information.