Everything depended on the assumption that the coolant had a backup system.Once that assumption was mooted by the tsunami, the flaws in the rest of the system became known.
Not really. The real assumption that failed was that even if there was a complete loss of power in the plant, power could be reasonably quickly (8 hours) provided from outside the plant. The problems escalated because no supplies were available due to tsunami devastation, not even freshwater. The power grid was so damaged that an extra cable had to be laid to get any external power.
One of which is that once you lose cooling and can't get it restarted, you will inexorably have to vent hydrogen into a closed space full of air. Another is that there is no way to vent it to the outside to reduce the effects of an explosion.
The hydrogen was vented inside the containment on purpose, to allow activation products to decay. It could be vented outside the containment, but this would increase the radiation emissions, which the operators desperately wanted to minimize at that point. Hydrogen explosion was deemed an acceptable risk. It looks like this kind of mindset, "reduce public radiation exposure at all cost", is what caused the situation to escalate.
Another is that if the cooling system is completely bunged, there's no way to throw external coolant on the thing that has any effect.
The design assumption was that once cooling completely fails, the reactor will be drained, sealed and allowed to melt down. But this would necessitate a very costly cleanup which TEPCO wanted to avoid.
And another is that they stored the "spent" fuel rods in bunches in what is basically an open swimming pool, so that any chance it gets to evaporate the water around it will result in a fire.
Storing them elsewhere would necessarily expose the workers to more radiation. The point of the temporary storage near the reactor is to allow the fuel to lose most of its radioactivity before it is moved to a longer-term storage location.
What's criminal here is that these things were known to be bad assumptions long ago, but these reactors were operating as originally installed.
Each of the design considerations had a lot of thought behind it. The real problem is that the nuclear safety regulations are not based on a realistic risk analysis, but on fantasies (e.g. child drinking maximally contaminated water for an entire year, or somebody eating exclusively spinach for an entire year). As a result, the operators focused minimizing public radiation exposure rather than on stabilizing the facility, which was actually counterproductive.