[Florsuzo]

Hmm...I'm not sure how it went from overheating (running hot) to replacing timing belt, etc. here.

If your timing belt broke it definately would stop. But if the vehicle were overheating and someone kept driving it you're looking at other issues besides a timing belt. I suppose it's possible the timing belt seized momentarily on a frozen water pump, jumped a few teeth on the belt, and stopped working. I would think though it would have made horrible noises as this happened.

FYI the water pump is driven by the timing belt.

Question you want to solve first (if possible) is why it started running hot.