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For a lifetime fix, drill out the stripped hole twice the diameter of the screw all the way through the wood backing, which would be 3/8" for a #10 screw, fill completely with White Marine-Tex, sand flush and re-buff the area. Then re-drill for the screw. The fitting will cover the 3/8" dia round, White Marine-Tex filling.

You don't even need to caulk this situation, since now the screw will be seated 100% in a Marine-Tex epoxy "plug". Marine-Tex holds screws like iron, forever!