It is explained in the gemara that if a nonJew slaughters an animal, it is as if the animal died by itself. This is because the non-Jew is not subject to the commandments, which include ritual slaughter. The Rambam explicitly states that even if the non-Jew slaughters under the supervision of a Jew using all the correct methods, it is still invalid.
Masechet Chullin 13a
The Mishnah on today’s daf teaches that if a non-Jew performs shechita
– ritual slaughter – on an animal it is not kosher.
Rabbi Hiyya the son of Rabbi Aba quotes Rabbi Yochanan as teaching
that the Mishnah should be understood as forbidding eating the meat of
an animal slaughtered by a non-Jew, but permitting its use for other
purposes, inasmuch as we do not assume that the non-Jew had intentions
to slaughter the animal for purposes of idolatry.
The reason is explained in Art Scroll Chulin, volume I, 13a3 Note 32 to the Mishnah at the bottom of the page, citing Tosfos, Rambam (Maimonides) Hilchos Shechitah 4:11 and Rashi. It is based on the Torah verse Deuteronymy 12:21 that the you reference means only someone subject to the commandments can slaughter.
Update Art Scroll gemara Chulin 13a3 Note 32:
The mishnah teaches that if an animal is slaughtered by a gentile, the
meat is neveilah, even if the slaughtering was supervised and
adhered fully to halachic procedure (Rashi)
A gentile's act of slaughter is not valid because the Torah states
Deuteronymy
12:21
... And you shall slaughter ... and you shall eat, which indicates
that what you may eat is slaughtered by you - i.e. a Jew who is
subject to the laws of eating meat only from a slaughtered animal
(Tosafos to 3b קסבר) Rambam Hilchos Shechitah
4:11
advances a different scriptural source for the disqualification of a
non-Jew's act of slaughter, but that source is challenged by the
Rosh Par. 5
Deuteronymy 12:21
כִּי יִרְחַק מִמְּךָ הַמָּקוֹם אֲשֶׁר יִבְחַר יְהֹוָה אֱלֹהֶיךָ
לָשׂוּם שְׁמוֹ שָׁם וְזָבַחְתָּ מִבְּקָרְךָ וּמִצֹּאנְךָ אֲשֶׁר נָתַן
יְהֹוָה לְךָ כַּאֲשֶׁר צִוִּיתִךָ וְאָכַלְתָּ בִּשְׁעָרֶיךָ בְּכֹל
אַוַּת נַפְשֶׁךָ:
If the place the Lord, your God, chooses to put His Name there, will
be distant from you, you may slaughter of your cattle and of your
sheep, which the Lord has given you, as I have commanded you, and you
may eat in your cities, according to every desire of your soul.
Rashi
you may slaughter… as I have commanded you: We learn [from here] that there is a commandment regarding slaughtering, how one must
slaughter. [Since this commandment is not written in the Torah we
deduce that] these are the laws of ritual slaughtering given orally to
Moses on [Mount] Sinai. — [Sifrei ; Chul. 28a]
Rambam Hilchos Shechitah 4:11
11 When a gentile slaughters, even though he slaughters in the
presence of a Jew, [using] a finely [honed] knife,23 and even if he
was a minor,24 his slaughter is a nevelah. According to Scriptural
Law, one is liable for lashes for partaking of it,25 as [implied by
Exodus 34:15]: "[Lest] he shall call you and you shall partake of his
slaughter." Since the Torah warns lest one partake of his slaughter,
you can infer that his slaughter is forbidden. He cannot be compared
to a Jew who does not know the laws of ritual slaughter
12 [Our Sages] established a great safeguard concerning this matter,
[decreeing] that even [an animal] slaughtered by a gentile who does
not serve false deities26 is a nevelah.27
Shmos 34:15
פֶּן תִּכְרֹת בְּרִית לְיוֹשֵׁב הָאָרֶץ וְזָנוּ | אַחֲרֵי אֱלֹהֵיהֶם
וְזָבְחוּ לֵאלֹהֵיהֶם וְקָרָא לְךָ וְאָכַלְתָּ מִזִּבְחוֹ:
Lest you form a covenant with the inhabitant[s] of the land, and they
[the gentiles] go astray after their gods, and they offer sacrifices
to their gods, and they invite you, and you eat of their slaughtering,
