I'd assume the design as proposed isn't overbuilt, so we'd need to go from, say, 4 months cycle times to 8 hour cycle times, which is maybe 350x.

While that's not going to increase the cost by 350x directly, it is going to change the character of the pile from a bunch of dirt to a bunch of dirty pipes. This makes a lot of the simplifying assumptions no longer work; like you can no longer ignore the heat losses through the rods, or the lower thermal mass of the rods.

And to be clear, you can do this. There are faster-cycling thermal storage solutions out there. It's just not implied from the claim that these solutions would be so much better than batteries.

The heat right next to the rods is easier to access. With less dirt per rod, you can also get more watts per rod, so the number of rods doesn't need to scale directly with the cycle time.