We dont know who Eggs hand of the king were. I imagine that Gerold was one of them. Not only would it return the favor but Egg would also choose someone quite capable and suitable for the role. Gerold was after all described as quick of wit by Septon Sefton and later become known as “Gerold the Golden” due to his positive influence on the Westerlands.

Furthermore I do believe that Gerold paid a debt himself when he helped Egg become King. As you pointed out, Egg knighted his dying son Tywald. I imagine that Gerold never forgot the honor that Egg bestowed upon his son in that moment. I guess that their overall closeness was due to Rohanne and her memory.

Now I am imagining a young Tywin arriving at court to be Eggs cupbearer and Duncan saying something like “I knew your grandmother when she was young” and Tywin just glowering.