Following real-world experience I am taking the position that only ground radials are capable of pushing a part 15 AM station's range to its maximum legal outreach.
So many of us have tended to believe that a ground rod can accomplish something of the same thing but I find otherwise.
Think of it as a 3-choice process...
A part 15 transmitter with only a 3-meter vertical antenna and no serious ground will not provide impressive reach. The relationship between antenna and ground is at best a very high impedance affair with little useful result.
A part 15 transmitter attached to a ground rod will tune-up beautifully on the output meter because the impedance between transmitter and ground is sufficiently reduced to provide a ground path to complete the circuit... except, that the highly resonant well tuned signal will be confined to an area near the ground rod and will not extend to the far field. In fact a ground rod is merely a negative dipole element buried in the ground where it loses potency.
Only ground radials will serve the radiated signal out to the distance by "claiming" the ground-plane and coupling it with the antenna system.
The final frontier amounts to trying different numbers of and lengths of radials.