I agree with most of your points, however: most of the accidents have more to do with flaws in regulations than mistakes from the nuclear engineers. Chernobyl, which is arguably the worst and most iconic nuclear disaster of all times was mostly caused by a severe violation of procedures.
When you take into account he amount of energy that has been produced by nuclear power since the 50s, I'd say we have a pretty good record when it comes to overall safety.
The biggest problem with Chernobyl was the reactor design, which had a positive void coefficient, which outside of something like CANDU reactors is considered insane today. It’s true that their experiment with simulated blackout was the proximal cause, but the fact that increased steam production led to higher power output was the big factor that made it such a huge disaster.
I believe the rods were graphite, just like the moderators. Of course the problem with that was that by the time they were inserted the reactor was already out of control. All the graphite rods did was burn, which certainly didn’t help matters! Worse, after a while the blocks of graphite used as the moderator also burned, and was difficult to extinguish. What they needed (other than better reactor design) was a ready store of neutron poisons that would have flooded the reactor vessel before rod insertion. Unfortunately, by the time they were using neutron poisons the explosion and fire had already destroyed anything like containment.
The biggest problem with Chernobyl was the reactor design, which had a positive void coefficient, which outside of something like CANDU reactors is considered insane today. It’s true that their experiment with simulated blackout was the proximal cause, but the fact that increased steam production led to higher power output was the big factor that made it such a huge disaster.