My case clock is full of pointy-toothed gears, as are most small clock mechanisms (including watches) I’ve looked at.
You might be correct in that large clocks (like in buildings and clock towers) use square-cut teeth, probably for strength and positive engagement.
I will put in a big plug for visiting it in Athens. The museum has a 3-4 room exhibit and exploration of why it was built, how it was used, and how they reconstructed it. But the showpiece is certainly the mechanism itself which pictures don’t really just for. You can see the gears and some of the minute engravings/markings. One can see how complicated it really is very easily- it isn’t simply a green corroded lump like the pictures show.
My pre-teens spent over an hour in the exhibit as it really was well done.
Just because someone made a machine doesn’t mean that it worked.
Most watches I’ve looked at have a variety of gear teeth types - some pointy, some not; most, if you look at them really closely, are rounded at the end.
The mechanism is remarkable in that all of its gears, for big wheels and small, have teeth that are equilateral triangles - easy to make with a file. Just take a round wheel, mark it off with equal divisions to the required number of gears, and file indentations to that pre-set depth, and you have your gears.
“Modern” 19th century gears were cut in a more complex manner.
The reconstructed devices do work. (There was a Scientific American article on the subject. Plus links in this thread.)
It is possible that there’s some “projection” going on, and that the reconstructions are tainted by a bit of wishful thinking. But there’s no actual evidence of that.
The best evidence is that the original device really did work.