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Shortly speaking, Saros cycle does not predict solar eclipses. It predicts Lunar eclipses, and not with 100% reliability. And the knowledge of Lunar eclipses does not require your position. They occur simultaneously for all observers, and can be used to determine longitude. Imitating the Saros cycle with a mechanical device is not difficult, one can copy the Antikythera mechanism after all. And I have actually seen working cipies advertised on the Internet.

Ancient Babylonians probably could use Saros to predict Lunar eclipses, and a device based on their methods can be constructed and this device will be simpler than Antikythera mechanism: I imagine a device resembling a slide rule.

Solar eclipses are much more difficult and the ancients could not predict them. (Contrary to what many popular writers and even historians write). This sentence probably requires some explanation: what does it exactly mean "to predict". Saros cycle permits to to determine the days when a solar eclipse MAY happen, but it does not mean that it will actually happen. And it does not tell you anything about the location where it will happen. To make exact and reliable predictions of solar eclipses, advanced theory of the Moon motion is required, the theory which was achieved only in 18 century, and I suppose that any mechanical device based on this theory will be impractical; the theory is too complicated for a mechanical device.

The commonly repeated story about Thales predicting a solar eclipse is interpreted by historians of astronomy that he predicted the year in which the eclipse will happen, and even this by sheer luck.