The bigger problem with Rods of God is that they just wouldn't work.
There are several reasons, let's take them one by one.
1. Kinetic energy is not that high compared to chemical energy, and is completely negligible compared to nuclear energy. For a kinetic(non-explosive) projectile to deliver the energy of its mass equivalent in TNT, it needs to travel at about 2.9km/s, or Mach 8.5. In other words, a rod weighing one ton hitting a target at 2.9km/s delivers a destructive energy equivalent to one ton of TNT. Since kinetic energy goes up with the square of the speed, such a rod traveling at the first orbital speed, 7km/s, would have a yield of about 5.8 tons TNT equivalent. For comparison, the largest non-nuclear bomb used in WW2 was the Grand Slam, with a yield of 6.5t TNT equiv, and the Hiroshima bomb had a yield of about 15000t TNT equiv.
2. Losses due to atmospheric drag. An orbital body can't magically hit the ground with its full orbital speed. It will lose a lot of its energy to air drag. The wikipedia page on the "Rods of God" [1] mentions a 2003 US Air Force study where the terminal velocity of these rods was Mach 10. That's only slightly higher than the Mach 8.5 I mentioned above, so the final conclusion was a bit underwhelming: "the practical applications of such a system are limited to those situations where its other characteristics provide a clear and decisive advantage—a conventional bomb/warhead of similar weight to the tungsten rod, delivered by conventional means, provides similar destructive capability and is far more practical and cost effective."
3. You can't simply drop something from orbit. A fun little experiment is this: you are a cosmonaut doing an EVA (extra-vehicular activity) and just for the fun of it you throw a small object towards Earth, at about 10m/s. What happens to the object? The cosmonaut's name was Leonov, who told the story to V.V. Beletskiy, who did the math, and the story is told beautifully by V.I. Arnold in [2]. The object goes initially towards Earth, but then starts turning, and does a full ellipse, and meets again the space station about 1.5 hour later, coming from above (well, technically it doesn't meet it, it has a deviation of a few dozen meters). The moral of this story is that you need some sort of launching mechanism to make the rod leave the satellite and go towards its target, maybe a rocket or some sort of a gun. If you want to deorbit something in the optimal way, you use a Hohmann transfer; I did a back of the envelope calculation and the propellant mass needed to deorbit is a bit less than 10% of the mass of the projectile. However, with a Hohmann transfer we are talking about more than an hour from the start of the strike to hitting the target. If you want to achieve minutes, you need to pay a price. I don't know how to do this calculation, maybe a KSP aficionado can help here, but it would not surprise me if the propellant ratio shot up to 30%. On top of that you need propellant to maintain the altitude of the satellite over the years. This adds mass and complexity to the satellite. Let's say you want to have a bundle of 20 Rods of Gods up there, 10t each. You get 200t, and then you add propellant, rocket engines, solar panels, etc, etc, I can't imagine you get away with less than 400t. For comparison the current mass of the ISS (the biggest satellite every built) is 370t and that cost so far $150BN.
There are several reasons, let's take them one by one.
1. Kinetic energy is not that high compared to chemical energy, and is completely negligible compared to nuclear energy. For a kinetic(non-explosive) projectile to deliver the energy of its mass equivalent in TNT, it needs to travel at about 2.9km/s, or Mach 8.5. In other words, a rod weighing one ton hitting a target at 2.9km/s delivers a destructive energy equivalent to one ton of TNT. Since kinetic energy goes up with the square of the speed, such a rod traveling at the first orbital speed, 7km/s, would have a yield of about 5.8 tons TNT equivalent. For comparison, the largest non-nuclear bomb used in WW2 was the Grand Slam, with a yield of 6.5t TNT equiv, and the Hiroshima bomb had a yield of about 15000t TNT equiv.
2. Losses due to atmospheric drag. An orbital body can't magically hit the ground with its full orbital speed. It will lose a lot of its energy to air drag. The wikipedia page on the "Rods of God" [1] mentions a 2003 US Air Force study where the terminal velocity of these rods was Mach 10. That's only slightly higher than the Mach 8.5 I mentioned above, so the final conclusion was a bit underwhelming: "the practical applications of such a system are limited to those situations where its other characteristics provide a clear and decisive advantage—a conventional bomb/warhead of similar weight to the tungsten rod, delivered by conventional means, provides similar destructive capability and is far more practical and cost effective."
3. You can't simply drop something from orbit. A fun little experiment is this: you are a cosmonaut doing an EVA (extra-vehicular activity) and just for the fun of it you throw a small object towards Earth, at about 10m/s. What happens to the object? The cosmonaut's name was Leonov, who told the story to V.V. Beletskiy, who did the math, and the story is told beautifully by V.I. Arnold in [2]. The object goes initially towards Earth, but then starts turning, and does a full ellipse, and meets again the space station about 1.5 hour later, coming from above (well, technically it doesn't meet it, it has a deviation of a few dozen meters). The moral of this story is that you need some sort of launching mechanism to make the rod leave the satellite and go towards its target, maybe a rocket or some sort of a gun. If you want to deorbit something in the optimal way, you use a Hohmann transfer; I did a back of the envelope calculation and the propellant mass needed to deorbit is a bit less than 10% of the mass of the projectile. However, with a Hohmann transfer we are talking about more than an hour from the start of the strike to hitting the target. If you want to achieve minutes, you need to pay a price. I don't know how to do this calculation, maybe a KSP aficionado can help here, but it would not surprise me if the propellant ratio shot up to 30%. On top of that you need propellant to maintain the altitude of the satellite over the years. This adds mass and complexity to the satellite. Let's say you want to have a bundle of 20 Rods of Gods up there, 10t each. You get 200t, and then you add propellant, rocket engines, solar panels, etc, etc, I can't imagine you get away with less than 400t. For comparison the current mass of the ISS (the biggest satellite every built) is 370t and that cost so far $150BN.
[1] https://en.wikipedia.org/wiki/Kinetic_bombardment [2] https://www.amazon.com/Mathematical-Understanding-Nature-Phe...