That one must be Jewish to shecht an animal is not necessarily obvious. One might have argued that as long as the meat is slaughtered properly, it matters little who actually did so. As we have noted, shechita itself is what we call a matir, as opposed to a mitzvah. It is a necessary prerequisite to allow us to eat meat but, as it is not an actual mitzvah to slaughter an animal, there is no inherent reason it can’t be done by anyone and everyone. It is for this reason that Gemara (Sukkah 8b) specifically permits a sukkah to be built by a non-Jew. Despite the fact that that building the sukkah is at the very least a hechsher mitzvah, a preparatory act enabling the performance of the mitzvah, there is no need for a Jew to build it. We must sit in one, but it matters little who builds it.
And while pretty much anyone can shecht an animal, there is one primary exception. “The shechita of an idol worshipper renders the animal a neveilah, an unslaughtered carcass, which imparts ritual impurity through carrying” (Chulin 13a).
The question the Gemara asks is not why the shechita of an idolater is invalid, but why the meat is “only” considered a neveilah, prohibited to eat, but permitted to derive benefit from. As a general rule, anything used in the worship of avodah zara, becomes prohibited to derive benefit from, in any and every form. We may not sell it, give it to a non-Jewish neighbour or even feed it to our pets, but must destroy it. The Gemara assumes that when an idolater slaughters meat, he has the service of his god(s) in mind. For a religious person, meat was (is) primarily a means to come closer to God, through being offered as a sacrifice or as a means to express thanksgiving to our Creator. Initially, Jews were only allowed to eat meat in the context of sacrifices, and only later did the Torah allow basar ta'avah, meat eaten literally “out of lust”. Jewish tradition after the destruction of the Temple viewed the eating of meat as a means of celebrating our religious festivals. Why, then, may we derive benefit from meat slaughtered by an idolater?
The Gemara’s answer is most instructive, even a little scary. Rav Nachman argues that “most of those who worship idols are not heretics, and this is in accordance with the view of Rabbi Ḥiyya bar Abba in the name of Rabbi Yoḥanan that, ‘gentiles outside the Land of Israel are not idol worshippers; rather, they are [only] following the customs of their fathers’” (Chulin 13b). People can go through the motions of idolatry, attend pagan temples, chant the prayers and perform the rituals. But if they are doing so only because they are following the customs of their parents, they are not truly idolaters. Faithful children, yes; but idolaters, no.
The Tosafists (Avodah Zara 2a s.v. assur) initially suggest this as the reason why the Mishnaic ruling forbidding Jews from doing business three days before the festivals of idolaters is ignored. The idolaters of France and Germany in the medieval period, Tosafot suggest, could not really be classified as idolaters because they were only following the custom of their parents. Tosafot reject this answer, not because it is not true but rather, because the Gemara rules we still may not conduct business with them on the day of the holiday (though we may do so on the three days beforehand). Even if the people were not idolaters, they were engaging in acts of idolatry.
But the niceties of idolaters and idolatry need not concern us.
What applies to idolaters equally applies to us. If it cannot be considered idol worship if one is just following in the ways of one’s parents, can one who follows mitzvot out of family tradition be considered an oved Hashem?
It is beautiful when one follows the traditions of one’s parents, but worshipping G-d requires much more. It requires, at a minimum, the realization that mitzvot are commands of the Divine Creator and we are to dedicate our life to carrying out His will—even, the halacha affirms, if it conflicts with the will of our parents. There is value in doing mitzvot even if we are not motivated to do them as such—just as there is meaning to idolatry even if it is only done out of habit—but a true oved Hashem yearns to carry out the will of G-d.
It is the notion that one is following “the customs of our parents” that serves as the basis as the ba’al teshuva movement. Already close to 900 years ago, the Rambam (Hilchot Mamrim 3:3) ruled that while we must bitterly fight and harbour extreme negativity towards the Karaites, this only applies to the founders of the movement. But their children, raised to be Karaites, cannot be “blamed” (and perhaps should be commended) for being Karaites. Thus, we must show them love, and may not, G-d forbid, harbour ill will towards them. Similarly, those brought up in non-observant homes are non-observant for good reason and to look down upon them is silly, wrong and sinful.
Yet if we are honest and consistent, the same holds true for those who are observant. How much can one take “credit” for, and how much is just a result of having been brought up that way? Perhaps it is for this reason that those who become observant are deemed to be on a higher religious level than even the most pious of people from observant homes. Only in cases such as theirs can we be sure that they are not just following the religious ways of their parents.
I think it is instructive—even if it’s impossible to really know—for those brought up in observant homes to ask themselves, had they been raised, say, in a secular home, do they think would have become observant? I know I would personally like to answer yes to this question, but the chances are very strong that the real answer is no. I love Torah and feel so fortunate to have been brought up in an observant home, and for that alone I am proud and thrilled to carry on “the customs of my parents”. But I know and realize that much of any good I may do would not have been possible without the efforts of my parents.
None of this means that we are not responsible for our actions and that we cannot go beyond our upbringing—for good or for not-so-good. But it does mean that we must be given leeway to make our own life choices. Responsible parents know they must leave room for their children to make important choices on their own, even as they struggle with knowing where to draw the necessary lines. Not only is such necessary to develop well-balanced children, it is necessary to develop children who are capable of worshipping G-d.
 It is in this respect akin to a get, which is a necessary prerequisite allowing the couple to remarry; but clearly, there is no mitzvah to get divorced. Yet the Torah itself requires the get be written lishma, for the specific couple, and Jewish law rules that only a Jew can create the conditions of lishma.
 While some who worship idols truly are heretics, as Rashi (Chulin 13b s.v.ein rov) notes, “we need not concern ourselves with the minority”!
 As Tosafot (Chulin 13b s.v. nochrim) points out, while in the Holy Land people are more religious, nonetheless, the influence (remember, we are talking 12th century here) of the Diaspora is such that even in Israel, “they are not so attached to idolatry that their actions [i.e., slaughtering meat] are done for the sake of idolatry”.
 Nonetheless, even if they not be idolaters, and their slaughter is not prohbited to derive beneift from, Jewish law does require that shechita be done by a Jew (Shulchan Aruch, Yoreh Deah 2:1).