The appliance nobody thinks about until the kitchen fills with smoke. Usually a motor, a capacitor or a duct that was never really connected.
A hood that runs but pulls nothing is either ducted into a dead end, blocked with grease at the damper, or fitted with filters that have not been washed in years. Aluminium mesh filters go through a dishwasher perfectly well and that alone restores a surprising amount of airflow.
A hood that hums and will not start is usually the capacitor. A hood that howls has a failing motor bearing. Neither is a big job, and on built-in and downdraft units it is far cheaper than replacing a hood that was specified to match the cabinetry.
Plenty of kitchens have a hood that was installed as recirculating, with a charcoal filter and no duct at all. It will never pull like a ducted one, and when someone says the hood stopped working, occasionally the honest answer is that it was never doing much to begin with.
Aluminium mesh filters, yes, in the dishwasher. Charcoal filters in recirculating hoods are replaced, not cleaned.
Usually the capacitor or the speed switch. Both are inexpensive parts.