This is an excellent question which touches on several important ideas discussed at length in the Torah in general and in Torat HaChassidut in particular.
The idea that Israel is the Chosen People is derived from the Torah of Moshe Rabbeinu, parshat v'Etchanan, chapter 7:6 which says:
כי עם קדוש אתה ליהוה אלהיך בך בחר יהוה אלהיך להיות לו לעם סגלה מכל העמים אשר על־פני האדמה
Many commonly translate Am Segulah as the Chosen People because commentaries like Ramban say we are given this title as a consequence of G-d choosing us at the time of the giving of the Torah. That this choosing is out of His love for Israel like is understood from the continuation of the concept in Sefer Devarim, parshat v"Etchanan 7:7-8.
But segulah is more accurately translated like is found in the Jastrow dictionary as precious or cherished or highly prized like is found in most classical commentaries like Ibn Ezra and the Tur HaAruch.
But at the same time, it is worth pointing out that the Alcalay dictionary translates segulah as adaptable. This would mean that because G-d chose Israel from among all nations, we became more flexible, meaning more adaptable than any other nation on earth to whatever changes would come to the world.
This idea speaks more to the ability of the Jewish people to survive, come what may, and to reproduce and raise children. This understanding is also supported by the concept of Survival of the fittest as espoused by Charles Darwin and Herbert Spencer found in modern science.
Such a perspective, which is according to the plain meaning (פשט) adds deep meaning to the phrase Torat Chaim, the Torah of Life, and that "It is a Tree of Life to all who grasp hold of it." And this is in keeping with the general teaching from Moshe Rabbeinu in Devarim 30:19 which states:
העידתי בכם היום את־השמים ואת־הארץ החיים והמות נתתי לפניך הברכה והקללה ובחרת בחיים למען תחיה אתה וזרעך
From a historical perspective, the simple fact that the Jewish people have survived intact from all the ancient races is one of the great, revealed miracles. Bestowing this extra level of adaptability, similar in effect to G-d's dispersing Israel all over the world (which looks on its face to be a hardship), is certainly a great kindness from the Creator which has served the Jewish people throughout the ages.
And if this idea of G-d choosing Israel is considered in all its details, it is understood that this is not speaking about the Jewish soul, which is eternal and which transcends the physical, material world, but to the physical, material body.
And this is stated explicitly by the Alter Rebbe (see the bottom of the page) in chapter 49 of section Likkutei Amarim in the Tanya.
And this leads to the question you raise that there appears to be a dual nature to the Jewish people. That from the external perspective, the Jewish body appears to be the same as that of the non-Jewish nations. But from the inner perspective, the Jewish soul is completely different from that of the non-Jewish nations. This distinction of the Jewish soul is expressed by Rabbi Meir Ibn Gabbai in Sefer Avodat HaKodesh, section 3 (Chelek HaTachlit), chapter 17 beginning with the words, "כי האדם הוא מה שהוא בשכל וכו".
Or to put it in the words used in your question, from the perspective of the Jewish soul, we are considered to be like a first born son to Our Father in Heaven. And like you point out, true choice can only be when there is no compelling distinction between those things being chosen.
But the Jewish people have two types of relationship with G-d (אבינו מלכנו), that of a son (בן) and also that of a servant (עבד). And the distinction between these two types of relationship is discussed in detail in the Chassidic discourse of the 5th Lubavitcher Rebbe, Rabbi Shalom Dov Ber from the year 5666, beginning with the words, "ומקנה רב כו".
And through understanding the nature of this dual relationship, it underlines that, as you point out, choosing the Jewish soul is not a true choice at all, because there is a compelling distinction according to the nature of the Jewish soul as contrasted with the non-Jewish soul. And if that is understood, and the choice is about the Jewish body, which appears similar to the non-Jewish body, then why is it distinguished?
And this is discussed at length by the Lubavitcher Rebbe in Sefer HaMa'amarim Meluket, volume 4, page 270, beginning with the words, "ויובן זה ע"פ המבואר במ"א בענין נשמה וגוף".
And with greater detail in Likkutei Sichot, volume 11, pp. 5-7, chapters 6 and 7.
And also in Torah Menachem Hitvadiyut, volume 2, in the Discourse, All Israel have a portion in the world to come, pg. 325, beginning with the words, "ומזה מובן".
That the essence of this choosing is something which completely transcends the nature of creation as a whole, both in regard to the One choosing (G-d, אין סוף עצמותו מהותו ), and in regard to the root of that being chosen (the Jewish body, הגוף דישראל).
That just as the mitzvot are from a higher root (Ratzon, Will) than the Torah (Chochmah, Wisdom, Intellect), meaning Torah comes to explain and to clarify about the Mitzvot. But the performance of the Mitzvot is the primary thing like is understood from Shemot 24:7 which says,
ויקח ספר הברית ויקרא באזני העם ויאמרו כל אשר־דבר יהוה נעשה ונשמע
So it is in regard to the Jewish body and the Jewish soul. That ultimately, in the time to come, the Jewish soul will be sustained from the revealed inner root of the Jewish body which is higher than the Jewish soul.