There is a critical difference between the Saros cycles and the prediction of eclipses through science's model of the Solar System: the Saros cycles is only based on the general idea that the combination of periodical events tends to give periodical compound events, while the model of our Solar System gives us a whole explanation of the phenomenon, complete with accurate predictions of several aspects of the eclipses.

Just to give an idea of compound periodical events, think about this one: the garbage truck passes by my house every 3 days and the milkman passes by every two. This means that every 6 days I will have both fresh milk and garbage collection, and the remarkable precision of this compound event makes me believe there is a connection between garbage and milk. I can amaze myself with the curious fact that I never receive both milk and garbage truck on two consecutive Mondays, not even on two Mondays of the same month! Of course, there is no connection here except for the fact that both have a very stable periodicity.

Saros cycles are exactly the same. The crossing of the Moon through a plane that is perpendicular to the Earth's orbit and crosses both the Sun and the Earth is periodical (every 28 days or so) and the crossing of the Moon's orbital plane with the Earth's orbital plane being a line that crosses the Sun is also a periodical event (twice a year), so the concurrent occurrence of both has to be a periodical event, which happens to be every 18 years or so. It is just as the case of the Milkman and the Garbage Truck.

While the Saros cycles only explain the combination of two periodical events, anyone can get a reasonably good astronomical telescope and check that eclipses happen if and only if the Sun and the Moon pass through the same location of the sky at exactly 12 hours or 0 hours of difference. In fact, with careful measurements of the location in the sky of the Sun and Moon you can predict if the eclipse will be total or partial, and you can predict the shape of the shadow on the Earth or the Moon. With the Saros cycles you can only predict that 18 years after one type of eclipse the same type will happen again (but not always).

Edit: the Saros cycles are about 18 years, 11 days, not 7 years.