I wouldn't exclude the Roman Empire, as they had lathes and factories. I don't think that is the point. Watches, and other small objects are really not the subject. When did watches start being engine turned, I have no idea. When did they become commonly engine turned, not until the mid 19th c. at least in America, no doubt earlier in Europe. The first engine turning to be used on domestic silver was probably in the factories of Mathew Boulton in the 18th c.. it is not because the Romans were somehow stupider, they just didn't have the impetus. Engine turning, much the same as a table for engraving, was a modern mass production technique. Why spend a lot of time, energy, invention, etc. to create a table when a leather sack works just as good for hand engraving.
Exactly the same thing can be said about turning, or spinning. Beakers were raised in the 18th c. then seamed by some, and turned,or spun, by others c. 1800, but turning was not used in the US until about 1850. Finally beakers were drop forged. Of course flatware was commonly drop forged before beakers, because the weight necessary was less. Does that mean silver wasn't turned before 1800, no because I have had turned pieces from the 17th c. But it does mean that it wasn't common, and if someone claims a goblet is raised c. 1850 in America (as one book on Southern silver did) they are wrong. By 1850 mass production came to America, and goblets were spun.
We could carry this to an extreme, who invented the punch card, I would have thought someone just before I went to university in 62, but they were used in Jaccard production over 100 years earlier. The point to be made is that there was a time that varied from country to country, from silversmith to silversmith, etc when engraving changed, the methods changed, the tools used, at least when we talk about the support of the object, changed. The technique in the first video, which I call wriggle work, was a cheap method of engraving. It was used by apprentices to decorate silver, like mass produced Indian Trade silver. It is a technique that fills the bill, cheaply, and quickly. It should really not be compared in quality to the hand engraving produced by 18th c. silversmiths.