My view is that glue never gets to the bottom of a splinter and if you open the splinter to get the glue to go in further then the splinter always goes further than the glue
.
In these cases, I rasp out a shallow scoop taking out the
entire splinter (a drop of low viscosity superglue helps to show if there is any remaining crack which need rasping out. stain or ink would prob' do the same). I then glue on a carefully shaped patch of matched wood. If the patch is roughed out to about the right thickness it will flex and ensure good mating surfaces.
This fix can still be over wrapped if required. If done well the patch is nigh on invisible.
Del