The actual sunnah in Islam governing this says, verbatim: “Let your beards grow and trim your mustaches.” That doesn’t mean to let the beard grow wild forever; al-Ghazālī advised, in The Book of the Mysteries of Purity, “the bigger the beard, the smaller the intellect.” Ha!
The shaving of the mustache is not called for, as in the above quote it says “trim,” not “shave.” But religious folks want to one-up each other as holier than thou, so they start exaggerating and pretty soon it’s razor time for the lip, while they grow wild hobo beards to make the look as imbalanced as possible. Because moderation is a vice and extremism is a virtue to a certain mindset. Never mind that the actual sunnah teaches moderation in all things.
Bit of an addition, this is not one of the things Muhammad said came from Allah, it was all Muhammad. And I believe most agree he did it primarily to make Muslims different in appearance than competing tribes who were either clean shaven or had big mustaches, not for reasons of religious purity.
Judging from photographs of 19th century “Prominences”, as the German word might be translated, it seems that the beard sans mustache was a fashionable look. I think it works pretty well for an oblong face such as Lincoln had, or maybe it’s just that I’m used to it in his case. Otherwise, I think it makes one look like a buckwheat cake, particularly if one keeps the full beard in the greying stage, and acquires the round face that often comes with middle-aged weight gain.
Coming from the US eastern seaboard near Philadelphia, I can at least address the Amish part of the question.
The Amish are, for all practical purposes, pacifists. As you might suspect, they arose in parts of what is now Germany. In that part of the world, mustaches were associated intimately with the army. Thus, as a sort of distinguishing feature/protest, Amish men do not wear mustaches. As far as I know, beards for married men are traditional at a minimum but may have some distant Biblical connection (I don’t have a concordance handy so I can’t cite the chapter and verse, but I recall mention of the emissaries of one of the kings of Israel being humiliated by an adversary, which involved having their beards shaved. Perhaps a beard was taken as a sign of Biblical maturity and wisdom and a lack of a beard was equated to the absence of those qualities.)
You draw an excellent distinction. Supposedly the Prophet took a dislike to lush mustachios because that was the style worn by Persians, Zoroastrians. The Sassanid Persian Empire being a regional-power rival to the nascent Islamic enterprise, it was a political branding. The actual Arabic phrase used in the sunnah for it is “qaṣṣ al-shawārib” which means ‘trimming the mustaches’, not shaving. It’s funny that the Arabic word for mustache is shārib, which literally means ‘drinker’. Because whatever you drink, it drinks too. Along the lines of calling it a soup strainer.
As already noted, up thread, the Mennonite/Amish prohibition on mustaches is directly tied to their pacifist teachings. At the time when followers of Menno Simons carved out their own denomination among the Anabaptists, refusing to wear a mustache was a visible sign that one would not enter military service as either a volunteer or a conscript.
The Late Ottoman/Early Republican Turkish scholar Köprülüzade published a paper in 1929 on “Influence of Turko-Mongol Shamanism on the Sufi Orders.” One of the things he claimed as evidence for Altai shamanist influence on Sufism was: shaven heads and walrus mustaches. LOL Above all it was the members of the elite forces called Janissaries, who were concomitantly dervishes of the Bektashi order, who cultivated the biggest mustaches of all.
IIRC this is related to Jewish religious law - specifically, not allowed to travel on the Sabbath. I read somewhere about a neighbourhood in New York where they would string a ribbon or some around the whole block, so that they could visit neighbours.
The rules like to get very picky - obviously going to other buildings in the same property, etc. is not “travel”, but going to another house is… unless they are part of the same “place”. Sort of like, pushing an elevator button is “work” but walking onto an elevator that automatically stops at every floor is not doing work. And so on…