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But a 13A fuse in the plug for, say, a faulty hair-drier that has a fault but not a dead short, could sit there all day having 20A pulled through it - and through the hair-drier.
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This is not a likely scenario, with modern appliances anyway. The plug fuse is primarily to protect the flex and plug in the event of a short-circuit fault. The appliance is required to have its own protection against overload or dangerous malfunction. In the case of the hairdryer, there will probably be two separate devices; a self-resetting thermal trip and a permanent, non-resettable thermal fuse. Any fault that causes the hairdryer to draw a dangerous current through its flex and plug will cause one or both of those to operate, as it would be dissipating much more heat than it is designed to do. Importantly, the prtoection is detecting the hazardous condition itself, namely overheating of the hairdryer, which can occur in other ways too such as restricted airflow. This makes is by far a better solution than relying on a fuse to sense the corresponding overcurrent.