A fan alone is not enough , you would need grease traps to prevent sending it up the chimney and powerful fan to extract.
You could drill thru from outside just above extractor fitted in chimney which negates the problems of using the chimney .
DIY kitchens have 5year guarantee which they honoured when some of my doors start to chip , paint specialist sent out and confirmed bad batch and were replaced .
Example only as we don’t know what the hinge was as they were binned .
The doors require no holes drilled for the clip in hinge .Up to the OP to search for what he wants by way of replacement.
This type of hinge , fits 35mm cup and expands into hole when clip pushed closed , no screws required in door . Hinge repair plate can be used to fit hinges anywhere on carcass regardless of original position,
If you have a cordless drill ? Lock chuck on screw and then reverse it out , or use grips or pliers to slowly remove ( or you risk snapping it).You can reuse plug without removing ,If no screw protruding a HSS bit can be use to drill it out or drill new hole a few mm away .