Mel Sutter here; have been repairing players now into 50 years.
Jeff Bridges had problem with loose screws in player action.  This
problem I have met; where possible, I drill a hole 90 degrees to the
hole in question to insert a maple dowel.  Piano hammer shanks are
handy if the size is appropriate (7/32") or as large as is allowable.
For a long screw, a second hole below the first one might be needed.

Then, re-drill the minor diameter for the screw.  A screw will go in
easier then if the screw threads are rubbed over a piece of candle wax.
If drilling at 90 degrees is not possible, drill a large enough hole in
the original screw hole for a maple dowel, glue it in with woodworkers
glue, then drill the minor diameter for the old screw.  Sometimes a
pressure clamp is appropriate on the sides to make sure the new "plug"
doesn't split the nearby wood.

Mel Sutter