By the way, generally a "Talmudist" means someone who studies the Talmud; the rabbis who wrote the Talmud are known as The Sages, Hazal (an acronym for "our sages of blessed memory"), or the Tannaim (those before the year 200) and Amoraim (from 200 to 500).
Okay, let's back up here.
The reading of Deuteronomy is a very nuanced one, which your translation is lacking. (In fact your question only arises because you're using a faulty translation.)
... do not give your daughter to [the non-Jew's] son, and his daughter do not take for your son. For he will make your son veer away from Me.
Wait -- why would "he" affect your son? Shouldn't it be she?
The Talmud observes that elsewhere in the Torah, "son" can also mean grandson; therefore it reads:
... do not give your daughter to a non-Jewish man, or your son to a non-Jewish woman. The non-Jewish father will make your maternal grandson veer away from G-d.
Okay then we're missing symmetry. What about "the non-Jewish mother will make your paternal grandson veer away from G-d"? That's it -- a child of a non-Jewish mother isn't called "your grandson."
That happens to be the reading the Talmud uses. But to answer your question more broadly, you're trying to ask a question of metaphysics, which honestly is none of our business. (The "different souls" thing is debatable, put mildly.)
The Written Torah puts down a system of laws, but it's often unclear at first reading what they mean. Literally? Would be vague, bizarre, and occasionally contradictory. However I feel like it? Doesn't seem right. According to whatever's morally in fad when I read it? That quickly becomes "however I feel like it." We don't know? If G-d cared so much about getting a message across, it would be strange that it could get lost like that.
Which leaves ... according to the oral traditions and interpretations of the People of the Book, as recorded in the Talmud.
So the following questions are all answered in the same way:
- who is a Jew?
- what does "an eye for an eye" mean?
- Exodus says to wear tefilin but what are they?
- Moses says "slaughter like I commanded you" but no more details are given in the Bible, how do I do it?
- Exodus lists a few of the ingredients for the incense, but then says "and other spices" - what are those?
We believe that the will of G-d is expressed as a legal system (itself a fascinating concept, Lord Sacks writes about this a lot) and that the interpretation recorded in the Talmud is binding.