In the section of Tuczay's paper entitled "The Automaton and the Genie in the Bottle", she covers the history of mechanical moving figures both in literature and in history, from the legendary Daedalus to the real-life Heron of Alexandria. She notes regarding the latter that:
Several inventions mentioned in his Pneumatica bear resemblance not only to the figure of the flying horse and other automatic devices featured in The Arabian Nights, but also to numerous other examples in Middle High German romances and epics. In both the Middle East and Far East, the manufacture of automata of various types was quite common at the royal courts, as demonstrated by the casual mention of such mechanised figures by
early travellers such as Friar Odoric and Marco Polo (Odoric of Pordenone 1966, 222).
The "flying horse" mentioned in the passage about their destruction indicating demonic origins is not the flying horse from the eponymous story which you're asking about, but instead the one from "The Third Kalandar's Tale", a different story which also features a flying horse.
So what about "The Story of the Magic Horse", or "The Ebony Horse", a different story altogether? In this tale, the horse does not seem to be driven by any magic or demonic influences: it is not a living flying horse, but one made out of wood, which gains or loses altitude according to the turning of two screws on its body. It's more akin to a hot-air balloon. Disregarding for a moment the weight of ebony, this is almost a plausible device to actually make in real life. From Richard Burton's translation:
Then he bade the slaves bring the horse before him and they did so; and, when the Prince saw it, it pleased him. So (being an accomplished cavalier) he mounted it forthright and struck its sides with the shovel-shaped stirrup-irons; but it stirred not and the King said to the Sage, "Go show him its movement, that he also may help thee to win thy wish." Now the Persian bore the Prince a grudge because he willed not he should have his sister; so he showed him the pin of ascent on the right side of the horse and saying to him, "Trill this," left him. Thereupon the Prince trilled the pin and lo! the horse forthwith soared with him high in ether, as it were a bird, and gave not overflying till it disappeared from men's espying, whereat the King was troubled and perplexed about his case and said to the Persian, "O sage, look how thou mayest make him descend." But he replied, "O my lord, I can do nothing, and thou wilt never see him again till Resurrection-day, for he, of his ignorance and pride, asked me not of the pin of descent and I forgot to acquaint him therewith." [...] the horse gave not over soaring with him till he drew near the sun, whereat he gave himself up for lost and saw death in the skies, and was confounded at his case, repenting him of having mounted the horse and saying to himself, "Verily, this was a device of the Sage to destroy me on account of my youngest sister; but there is no Majesty and there is no Might save in Allah, the Glorious, the Great! I am lost without recourse; but I wonder, did not he who made the ascent-pin make also a descent-pin?" Now he was a man of wit and knowledge and intelligence; so he fell to feeling all the parts of the horse, but saw nothing save a screw, like a cock's head, on its right shoulder and the like on the left, when quoth he to himself, "I see no sign save these things like buttons." Presently he turned the right-hand pin, whereupon the horse flew heavenwards with increased speed. So he left it and looking at the sinister shoulder and finding another pin, he wound it up and immediately the steed's upwards motion slowed and ceased and it began to descend, little by little, towards the face of the earth, while the rider became yet more cautious and careful of his life. [...] And when he saw this and knew the uses of the horse, his heart was filled with joy and gladness and he thanked Almighty Allah for that He had deigned deliver him from destruction. Then he began to turn the horse's head whithersoever he would, making it rise and fall at pleasure, till he had gotten complete mastery over its every movement.
This is not the behaviour of a living horse, or one whose motion is driven by demons. It's simply a machine that one needs to learn how to operate.
Furthermore, there are no instances in this story where speaking the name of Allah causes the horse or any other machine/automaton to stop working. It does get destroyed at the end of the story, but only by the King literally breaking it into pieces.
The only connection between this horse and "evil" is that the Persian sage who created it is a deeply unpleasant figure. Because Prince Kamar wanted to protect his sister from marrying him, and their father the King imprisoned him, this sage decides to get revenge by attempting to rape Kamar's bride-to-be Shams. But there is no suggestion that his unpleasantness comes of being ruled by demons, or of any religious differences. He's just a bad person, albeit one who invented a marvellous thing.
So to answer your question:
How is this observation relevant in the context of “The Story of the Magic Horse” specifically?
I would conclude that it is not especially relevant. The ebony horse is not a thing of evil, nor is it destroyed by religious invocations, nor indeed does it bring about any evil events (Kamal almost gets lost with it, until he figures out how to use it, but even that is ultimately a good thing as it leads to him meeting Shams), even though its maker does.