I've seen many solid lead plugs originally done by Fox/Philly along with those in several Winchester Mod 12 Heavy Duck pump guns, and one just recently in a Baker SBT gun. In all of those guns the plug was at the back end of the stock, just a little under the buttplate or pad where the wood is quite thick. One Fox plug I measured, but didn't pull, was about 5/8" in diameter, and the Baker plug which was pulled is .520" diameter which suggests it was pressed lightly into a 1/2" drill hole. The long surface of the Baker plug is somewhat rough and looks like it was cast that way before pressing into the stock, to positively grip when under recoil set-back. All these many guns were done by the respective factories and none had cracked stocks or even the start of a crack. Yes, the exposed end of the plug was oxidized gray/white but that was easily cleaned if desired with something like a pencil eraser. Of course any gun with a lead plug what was done post-factory is subject to how that plug was done by whoever did it.
I can understand a stock splitting if someone hammers a way oversize plug into a hole, or let's say if someone pours molten lead into the hole.
Regarding using lead shot for recoil reduction or to help balance the gun, again shot will oxidize over a long period of time and when exposed to moisture, and probably the lead will also react with tannin in the wood. But as metals go, lead is relatively inert = increasingly less reactive than iron, nickle and tin, and a little more reactive than copper. Oxides will form over time but why wouldn't they creep into the open spaces between the shot pellets instead of creating pressure enough to crack the wood? Just based on the one picture provided here it looks like the crack is much farther forward than the solid lead plugs in the gun examples I cited. Maybe the hole drifted to one side while drilling, the wood "wall" was really thin there, and the crack resulted from release of internal stresses in the wood and/or from getting wet and drying out multiple times? All this is JMO of course.