Ok so at sea level the speed of sound is ~343 m/s, and these rods are theorised to impact at ~MACH 10, which is ~3.5 km/s, well short of ionisation velocity if we're in need of 7 km/s, which is ~MACH 20 at sea level.

Would a tungsten rod shed particles on its descent? Could it be coated to reduce particle loss?

If the only problem with this theory is the cost, well I wouldn't rule out superior propulsion technology than is known to exist. Granted, it's still pseudoscience, but I can at least entertain the idea that John Hutchison was onto something with his bowling ball experiments.