If you sink the hole much deeper, you run out of thread on the screw! As a bit of a bodge or bubba-fix, the lazy Armourers would drop a COLLAR, pin axis backsight into the hole to take up the slack. It works - until the next time. What I would say is to just do what we would do. Bore ONE SIDE of the deep hole out and insert/glue a wood plug. Then using the OTHER side as a guide, drill he centre hole through. When that'\s done, repeat with the other side, using the repaired side as the drill guide, There, done, properly once and for all..... Done hundreds of 'em

Do it properly, do it once I say and was taught