First, the idea that Aztecs literally thought the sun would stop if they ended sacrifices is about as true as Ancient Greeks thinking the rain would cease if they didn't keep up their sacrifices to Zeus. It's a theologically useful post-hoc rationalization if it does occur, but religion is not so simplistically transactional.

Second, the Sun had, in "fact," been destroyed four times previously. Hence the name for the age in which the Aztecs lived. the Fifth Sun. The most famous account of the eschatological cosmology of the Aztecs is the "Legend of the Suns" from the Codex Chimalpopoca. It describes how the four previous ages had ended in disaster, thus leading to the Gods restarting to world for our current age, which will end in earthquakes. None of this is more or less improbable than a war between Titans and Gods or any other mytho-religious explanations for the existence and/or current state of the world. The unbelievability of the stories stems from knowledge we are privileged to have today, not from defects in the thinking of those in the past.

To return to the idea of "blood makes the Sun run" as an oversimplification, we must keep in mind that human sacrifice was only the most visible and dramatic of an intricate web of religious rituals, and that a great deal of "sacrifice" of effigies, animals, and the "auto-sacrifice" of the worshipers piercing and nicking themselves to shed their own blood. The rites practiced by the Aztecs were not mere grease upon cosmic wheels, they were both management of the cosmos and proper ways of living, in addition to being a fulfillment of the moral debt owed by humanity to the gods who sacrificed themselves to create the Fifth Sun. Read in her 1998, Time and Sacrifice in the Aztec Cosmos, has a great summary on the idea:

Many scholars also have noted a close sacrificial bond between deat and destruction and life and creation, a bond I suggest correlates with Mexica concepts of transformation. Both the fact that one cannot live without eating (an act that necessarily destroys things) and the idea that all things are timed fit nicely together, for things must continually be destroyed if life's transformative processes are to go on; old things are "eaten" to create new things. Just as transformative reality fused time inseparably with things of space, so too a hungry cosmos that ate to live fused life and creation inseparably with death and destruction. (p.128)

The universe, in other words, is like a living entity or ecosystem, which, like all living things, is never static. Reality is an ever-shifting and adjusting thing, and Aztec religion and philosophy was focused on maintaining personal, national, and cosmological balance. So it's a bit more complicated than thinking the Sun would stop moving across the sky (besides, it'll be earthquakes, eventually).

PS - James Maffie is doing the best work on Aztec philosophy since Leon-Portilla. His entry on Aztec Philosophy is an excellent primer.