> Are nukes really as important as everybody thought they were after WWII?

Possibly moreso. Nuclear sovereignty is demonstrably above the conventional type. At the end of the day, having a forum where nuclear powers with long-range delivery capability can veto things reduces the risk of them using that capability to veto in the real world.

By the range requirement, Tel Aviv and Pyongyang qualify for UNSC inclusion. New Delhi and Islamabad do not—they will mostly just nuke each other.

I haven’t run the stats on this but it seems to me that countries which have nukes are more likely to invade other countries and then use the nukes as shields to prevent retaliation.

Out of the 9 current nuclear armed countries 5 have invaded another countries this century, and three of the most prolific invaders this century (Israel, USA, and Russia; each with over 3 invasions this century) are all nuclear armed. Out of the 4 countries which haven‘t invaded another country this century, two (India and Pakistan) regularly engage in border skirmishes and bombing campaigns. This leaves China and North Korea as the only two nuclear armed countries (out of 9) which don‘t regularly engage in foreign wars.

By our current experience, the proliferation of nuclear armed states is almost certain to end in a disaster at a previously unseen scale. We should be doing everything in our power to prevent this world of future horrors.