The timing belt is one of the most important parts of an engine. It makes the camshafts turn in time with the crankshaft, so without it an engine wouldn’t run. Some engines are what’s known as an interference engine, which means that when the valves open they take up space that the piston will take up when it’s at the top of its stroke, and this means that if the timing belt breaks, your engine is going to run into itself and destroy itself.
This car’s engine is luckily not an interference engine, which is good because the timing belt is falling to pieces. The engine would crank, but obviously not start. Some engine have timing chains, which rarely break or need replacement. Usually belts are recommended to be replaced every 100,000 miles or so to prevent destroying the engine.
You can see how bad the timing belt was on this car. Those stringy pieces on the floor are also parts of the timing belt.
Are there any drawbacks to having a chain rather than a belt?
A couple drawbacks are that they can be a little noisier, and they require lubrication, so there has to be oil supplied to it, while a belt can be damaged by oil.
Chains can break if put under a heavy load when very cold. I’m sure belts can too.