While I'm late to this discussion, I'd like to straighten out and clarify the issues involved. There are a number of matters here:
1) Yiras Shamayim. This requirement in a shochet predates chasidism by a long time, and is quite mainstream. Here is a quote from Shulchan Aruch (YD 18:17):
טבח שלא הראה סכינו לחכם (ונמצאת יפה), היו מנדין אותו. ואם נמצאת פגומה, מנדין אותו ומעבירין אותו. ובנמצאת יפה, יכול החכם למחול ואין צריך לנדותו. והאידנא נהגו למנות אנשים ידועים על השחיטה והבדיקה, ולהם מחלו חכמים כבודם, כי הם זהירים וזריזים, והרבה צריך ישוב הדעת ויראת שמים לבדיקת הסכין, הלא תראה כי יבדוק אדם פעמים שלש ולא ירגיש בפגימה דקה ואחר כך ימצאנה כי הכין לבו באחרונה, ובחינת חוש המישוש כפי כוונת הלב.
See also here.
2) So-called chasidic shechita. This originally was a question of forged vs. molten steel. Rabbis in the new chasidic community maintained that forged steel was unfit and that molten steel was superior, while mitnagdim and their Rabbis took great offense at the suggestion. Thus, chasidim had to rely on known chasidic shochtim if they wanted to adhere to the chasidic standard. This detail is no longer relevant as this standard has long become mainstream. See here for a historic overview.
3) Avoiding non-chasidic shechita. Interestingly, the baal ha'Tanya, probably the first authority to go on record and advocate extensively for the so-called chasidic knives, categorically asserts that it would be unthinkable to invalidate 'traditional' shechita on this basis (see previous link):
אם לפעמים מסובים בסעודת מצוה עם אנשי עירם, חלילה לפרוש מהם להחזיקם
כאוכלי נבלות ח"ו, הס מלהזכיר, ומעודי לא נזהרתי מהכלים אף מבני יומן.
מ"ש כבוד תורתם שנאמר להם בשמי על שחיטה שלהם שאינה בכשרות ח"ו, חלילה
חלילה לי להוציא דבה על עם ה' רבבות אלפי ישראל, ונאמן עלי אבא שבשמים,
וגם יעידו עלי כל המקורבים אלי, כי אינני נזהר מעולם מכלים אפילו הם בני
Of course, note the emphasis in the first letter on the 'occasional'.
4) Sticking to one's community. I imagine the issue here is not specifically a chasidic one at all. To oversimplify: communities used to be defined by geography. Jews living in X were X, those living in Y were Y. Every city had its demographics, its Rabbinate, and its shechita. With the advent of chasidism certain cities may have split in half, but the geographic boundaries persisted. This meant that a given city's shechita (or shechitos) belonged to, and was controlled by, solely itself/ves. If someone from 'out of town' would attempt to encroach on any community's shechita, he would be deemed an interloper, a מסיג גבול, and be excommunicated and run out of town. Today, many Jews, especially chasidim, still define themselves by European geography, and organize their communities in similar fashion. Thus, sticking to one's own community's shechita is more of an expression of a protective, exclusionary sentiment than a purely religious concern about kashrut.
ADDITION: 5) Difference between shechita and ordinary hashgacha. Here are my thoughts: a) The notion of Chasidic/community kashrus originated in Europe when the only kashrus job was shechita; applying the yiras shamayim/community argument here should necessitate similar standards for hashgacha, but perhaps this is an example of the original logic being abandoned and forgotten when the reality changes. b1) In terms of yiras shamayim, Shechita is the only instance where the act itself establishes the kashrus, whereas one could argue that all other areas of hashgacha are more about ensuring the lack of something unkosher occurring; those spiritually inclined could therefore argue that the impact of the shochet on the souls of those eating his food is profound to a much greater degree than anyone else involved in food production. b2) The laws arising every day in the slaughterhouse are more complex and costly than those typically arising in other areas of food production. b3) There may be certain chumros unique to shechita unique to each community as opposed to other areas of kashrus. c) Chasidic communities large enough to maintain their own shechita usually operate their own hechsher as well, and thus it’s possible that these communities do in fact tend to rely on their own kashrus where essential foodstuffs are concerned. d) Operating purely based on the ‘community’ explanation, one could argue that the same right that grants each community the power to preserve the strength of its shechita, allows for the relaxation of that power should they so desire it; thus, these communities apparently decided to maintain their grasp on shechita but willfully chose to allow for other kashrus powers to overlap with theirs where practically necessitated.