Earlier quoted context omitted.
It seems like you should build a water reservoir at a higher elevation than the core and then apply a similar principle where valves regulate the water stream, but if the valves lose power they fail open. The reservoir can be built so that there is always enough water to cool the core. For light water reactors this basically just amounts to a large pool up a nearby hill or in a water tower.
That is easier said than done - modern reactors are in the 1000 MW+ electrical power range, which means about 3x as much heat needs to be generated to get this much electricity - say 3000 MW. Even when you correctly shut down the chain reaction in the reactor (which correctly happened in the affected Fukushima powerplant) a significant amount of heat will still be generated in the reactor core for days or even weeks…
Some napkin math based upon heat capacity of water and assuming a 20 degree celsius input and 80 degree celsius output and 30MW heat results in about 120 liters per second of water flow needed. That is about 10 million liters of water per day, or about 4 olympic sized swimming pools. I don’t know how long you need to keep cooling for, but 10 million liters of water per day seems not insane and within the realm of possibility.
If you allow the water to turn into superheated steam you can extract much larger amounts of heat off the reactor as well.