If you had a solid wood cabin top, and screws large and plentiful enough to make it secure by screwing into that, you could clean out the old screw hols, epoxy fill them, and reset it with the screws again.
But personally I'd rather through-bolt them, either into a backing plate or into one of the matching interior handholds--if that was a good location for them.
Throughbolting should not be any cause for concern (leaks, etc.) if it is done properly, with epoxy sealing the edges of the holes, and a good sealant applied before setting the bolts. Consider that a 200# crew, holding onto that handrail, can put over a thousand pounds of shock load on it if the boat hits a rock, etc. and the crew is "thrown".
Still trust the screws that pulled out once before?