It's an extremely stupid idea. Your whole water column is going to be contaminated with fission products. And you won't be able to get any reasonable amount of power out of that contraption.
And even if you are stupid enough to actually do this, the fuel efficiency will be terrible. Your only negative feedback for fission is the Doppler effect and thermal expansion. So you will only be able to utilize a tiny percentage of the fissionable materials.
Yes, it would. Fuel is around 2-4% of the total costs for a regular nuclear power plant, but that's because regular reactors can burn it deeply. This reactor will only burn a couple of percents of the available fuel, so the fuel costs will probably be around 10-20 times higher.
BTW, this tradeoff can be acceptable for some very specific applications. Kilopower ( https://en.wikipedia.org/wiki/Kilopower ) is designed to use passive regulation.
Isolation from _what_? If you reactor melts down, it will contaminate the water column above it. And by having it inside the shaft, you won't be able to do any maintenance on it.
It's a stupid idea designed to filter out investors who are stupid enough to fall for it.
It boils off the coolant, creating a vapor lock, then water-zirconium reaction happens. Then it explodes and ruins the shaft. So you'll have to drill a rescue well, and then defuel the reactor to avoid any possible contamination.
Again, this is a monumentally stupid idea for no reason whatsoever.
It boils off a 1 mile deep water column? And you don't need a core catcher when you are already 1 mile deep in bedrock. It just goes to the bottom of the hole.
So this thing is just boiling off water from the mile deep water column, and nobody things "hey maybe we could add some more water"? How does a shaft "explode"? What is generating the explosive force?
And even if you are stupid enough to actually do this, the fuel efficiency will be terrible. Your only negative feedback for fission is the Doppler effect and thermal expansion. So you will only be able to utilize a tiny percentage of the fissionable materials.