Just like the title says, I watched this video https://youtu.be/CHMY4G9gTPA and then became intrigued.
An eggshell is a complex structure. From Wikipedia:
Boiling the egg removes the waxy cuticle from the outside of the egg, dissolves a small but not insignificant amount of calcium carbonate from the shell, damages the protein matrix that holds calcium carbonate crystals in place in the shell, and can disrupt or destroy the two shell membranes. All of these contribute to allowing bacteria a means to enter the egg and spoil it.
Note: Though calcium carbonate is actually less soluble in hot water vs cold, the rate of dissolution is substantially greater. Unless the egg is boiled in a saturated solution of calcium carbonate, it will lose material to the solution when boiled, and any pores in the shell become enlarged as a result.
In addition to Jason answer: egg is a live organism with repair and immunity capabilities. Small punctures in egg membrane may be healed, and bacteria are most likely to get killed by white blood cells using antibodies or other techniques. Boiling kills everything, including egg immunity system - existing antibodies are denaturated (they are protein too) and there are no white cells to produce fresh ones.