Lubavicher shechitah even where there is another perfectly mehudar chareidishe/chassidishe shechitah because only people who learn chabad chassidus can be yirei Shamayim.
I’ve looked through the Rebbe and Frierdiker Rebbe’s sichos, and asked around, and couldn’t find a single source where they said to eat only Lubavitch meat. And, by the way, far from everyone keeps Lubavitch Shechita. I know a _lot_ of people who eat Chassidisher Shechita.
They just said that meat has to come from a Yerei Shamaim, and if they come from backgrounds where beards are required, they must grow them (I’ve seen a letter of the Rebbe where he said that if a Shochet comes from a community where people traditionally shaved, shaving isn’t a sign of a lack of Yiras Shamayim). They don’t say anything about what, or where, they should learn, or to which community they should belong.
My suspicion where this custom came from, is:
1. It says that a Shochet has to have a lot of Yiras Shamayim. So since there’s no externally visible objective definition of “Yiras Shamayim”, people used the Shochtim they personally knew and trusted, and since people hung around their own community (Lubavitchers would daven in the Chabad shul and they would hold farbrengens together, while the non-Lubavitchers would obviously daven in their shuls and hang out together), many ended up eating only Lubavitch meat, and that’s how the custom developed. And even nowadays, Lubavitch is small enough that many people know who their Shochtim are personally.
2. Not sure about this one, but in the past (in Lubavitch circles), the Shochet of the town was a highly respected position, where they would frequently double up as the town Mashpia/Mashgiach, and the position would be close to that of the Rov. Was it this way in non-Chabad circles? Because if it wasn’t, then the Lubavitch Shochtim would have been (on average[1]) more Yirei Shamayim since a non-Chassid who was Frum and learned would prefer to work as a Rov, Mashgiach, Rosh Yeshiva, Maggid Shiur, or learn in Kollel. I’m not sure if this is true, but I very rarely hear stories of Litvishe Shochtim, while I do hear stories of Litvishe Rabbonim, Lubavitch Rabbonim, or Lubavitch Shochtim.
But my point stands either way. We consider our derech to be more Emes, but everyone else has a good derech and are fine and good Jews, and are Bichlal Amisecha. And no, we don’t consider your meat not-kosher in any way, and even those who keep Lubavitch Shechita, consider it just a chumra, and will eat Beliyos from other meats.
[1]. What I mean is that if, say, a Reb Chaim Shmulevich would work as a Shochet and the Lubavitch equivalent would work as a Shochet, we’d trust both equally. But since people like Reb Chaim would find jobs in Yeshivas, the Shochtim would come from the more mediocre bochurim. So if there would be a Lubavitch Reb Chaim vs a non-Lubavitch Shmerel, we’d obviously trust the Lubavitch Reb Chaim more.