Why are children who have a Jewish Father and gentile mother not considered Jewish?
The principle derives from the Mishna, Kiddushin 3:12. There it gives four different examples of possible sexual unions and relates the status of the child in each example:
כל מקום שיש קדושין ואין עברה הולד הולך אחר הזכר. ואיזו? זו כהנת לויה וישראלית שנשאו לכהן וללוי ולישראל. וכל מקום שיש קדושין ויש עברה הולד הולך אחר הפגום. ואיזו? זו אלמנה לבהן גדול, גרושה וחלוצה לכהן הדיוט, ממזרת ונתינה לישראל, בת ישראל לממזר ולנתין. וכל מי שאין לה עליו קדושין אבל יש לה על אחרים קדושין הולד ממזר. ואיזה? זה הבא על אחת מכל העריות שבתורה. וכל מי שאין לה לא עליו ולא על אחרים קדושין הולד כמותה. ואיזה? זה ולד שפחה ונכרית
In all instances in which there is an halakhic union (qiddushin) and no transgression is involved, the offspring follows [the status of] the father. What would this be? The daughter of a priest, a levite or an Israelite who marries a priest, a levite or an Israelite man.
In all instances in which there is an halakhic union but a transgression is involved, the offspring follows the defective status [ie: whichever of the two parents is lower in status than the other]. What would this be? A widow who marries the high priest, a divorced woman or a chalutzah (a childless woman whose husband died and who did not subsequently marry her brother-in-law) who marries a regular priest, a mamzeret or a temple servant who marries an Israelite, or the daughter of an Israelite who marries a mamzer or a temple servant.
In all cases in which there is not an halakhic union with this particular man but she could have an halakhic union with other men, the offspring is a mamzer. What would this be? One who transgresses any of the sexual crimes of the Torah [ie: incest, adultery, etc].
In all cases in which she cannot have an halakhic union with anybody, the offspring follows her [in status]. What would this be? The offspring of a slave woman or a gentile.
- Mishna, Kiddushin 3:12
This is the earliest formulation of what has come to be known as the matrilineal principle: in the event that a woman cannot contract an halakhic marriage with the man whom she has had sexual intercourse with, or any other man, the child follows her in status. Given that one of the two examples of such a woman is a gentile, the practical upshot is that the offspring of a non-Jewish woman is always non-Jewish, irrespective of who her partner is.
(There have always been dissenting voices to this principle, an opinion to the contrary even being recorded in the post-Mishnaic Avot deRebi Natan, but the halakha is as I have outlined it. For further information on the development of this principle and its potential social origins, I recommend Shaye J.D. Cohen, The Beginnings of Jewishness: Boundaries, Varieties, Uncertainties (Berkeley: University of California Press, 1999).)
Well basically, the Torah says Jewishness is determined matrilinearly, so that's all there is to it. In fact, the product of a Jewish mother and non-Jewish father is considered by Jewish law to technically have no father. (They are identified in a ketubah, for instance, as "the son of [mother's name]", or "the son of [maternal grandfather's name]." A kid whose father is a Jewish product of adultery is a lot worse-off in Jewish law than a kid whose father wasn't Jewish (in which case dad's parentage is totally moot).
I don't know where you got this "sins to the soul" business, but it doesn't fit with the laws described in the above paragraph, nor sound like anything I've heard in classical Jewish sources.
If you wanted to propose some basic suggestion as to understanding why (not "oh the reason is x and therefore x doesn't apply so I don't have to keep it"), it may be that mom tends to be more of an influence on kids than dad. Note that the Biblical source for matrilineal descent occurs in a warning on pagan influences -- "don't marry your son or daughter to a pagan, as the male pagan will influence your grandchildren" -- implying that the female pagan's children aren't even "yours", Jewish grandpa.