Here is how I'd do the repair;
Drill out the screw hole to about 1/2 inch
Cut a plug that size, or very slightly larger, from hardwood, across the grain
Apply resin glue to the plug hole and hammer the plug in
Cut a piece of metal about 2 inches long by 3/4 inch wide
Drill 3 holes in the metal, along the length, 1 in the middle and 1 at each end. Make the holes just big enuff for screws to clear
Place the metal over the plug. Centre it. Screw in screws of at least 1 inch length in the ends and drill a hole in the plug for your fitting
If the metal can be on the surface use round head screws. If the metal has to be flush. chip out the area for the metal to be recessed and use counter-sunk screws