Neutral is bonded to ground in one place only on an NEC compliant residential/commercial electrical system. This single bond is to ensure that any faults will clear when either the current from line-to-neutral via ground is too high in a non-GFCI circuit (overcurrent), or by ground fault (rating of the GFCI breaker 6-20mA depending on application) when GFCI is applicable under NEC.
The system ground rod is intended to pull the neutral line of the secondary side of the power company's transformer to a safe local level - without any ground, the secondary side
could float to a voltage where line-neutral is 120V, but line/neutral to earth upwards of >1kV (zap) as air moves over the wires giving them a charge (just like ragging your feet in wool socks over a carpet).
This is completely separate from a lightning ground system... That requires being bonded to the system ground to prevent lightning from jumping into the home's power system, but it also requires its own ground array and conductors separate from the rest of the electrical system ground.
Having many ground rods on a system ground does not "hurt" the fault clearing effect as the earth does not create a meaningful effect on the power circuit, since the circuit is completed from line to neutral, not line to ground. The only circuits that are completed from earth to an electrical system is ambient air charge (static/lightning) and RF. If you have your generator any appreciable distance from your house and its system ground, an extra ground rod at the generator will help with dissipating lightning strikes on the generator, near the generator.
If you're worried about multiple ground rods on your home's system, think about this - each of your neighbors are also required to have a system ground rod and neutral bond at their main point of entry under NEC, and power poles will have a ground rod to neutral every so often. The system is already done this way as the power company does not distribute "ground", only power