Suppose an engine belt breaks while you're off on a leisurely drive through the country -- or even while you're fighting your way through bumper-to-bumper traffic on the route to work. Your air conditioner will stop refrigerating the car's interior, your steering will become sluggish and your coolant system will stop working. Not a good situation and it happens fairly often. Engine belts are a weak link in the chain of a car's engine. The flexible materials that belts are made from are subject to fatigue and have a limited lifespan, often more limited than the lifespan of the engine itself. Belts more than a few years old can simply snap -- and they always seem to do it when it's as inconvenient as possible. Even if you look for cracks in the belt (a sign of wear) and have the belts replaced in advance you'll need to put your car in the shop and shell out a hefty chunk of change before you can drive it again. There are some adjustments that the owner can make to a serpentine belt using the belt tensioner, but most people would be happy if they never had to fool around with it.
The belts in beltless engines never become badly adjusted or break because, well, they don't exist. That makes beltless engines more reliable than conventional engines and reliability is a feature in a car that we can all get behind.