From a geopolitical strategic perspective, the goal of nation-states is not in fact to maximise chaos. You might care to revisit your premises and/or understanding.

The energy released even by nuclear weapons relative to that needed to melt or dislodge icecaps is puny.

The largest nuclear weapon ever tested, the Tsar Bomba, had a yield of 50 megatons TNT.

That's the equivalent energy of about 35 million barrels of oil. That's about 1/3 the amount of oil consumed worldwid per day.

There are a total of about 10,000 nuclear weapons worldwide. I'm not sure what the average yield is, though most US weapons seem to be around 500 kilotons nominal, so I'll use that, adjust your recipe to suite....

That gives 3.4 billion barrels of oil energy equivalent, or roughly 100 days' worth of global petroleum consumption.

Nuclear weapons are awesome, in the original sense, compared to traditional chemical weapons, for use in warfare. As a net total store of energy ... what humans expend in our normal course of life is expended somewhat more slowly, but at tremendous scale. The original Hiroshima bomb, for example released the same energy as a large electrical power plant generates in about 18 hours.[1] The bomb just releases it much faster (in a few microseconds) and in different forms (blast, heat, and radiation, rather than electrical potential).

Or, put another way: we're already nuking the poles, just somewhat more slowly and consistently than you suggest.

Sources:

Conversion calculations using GNU Units <https://www.gnu.org/software/units/>

<https://www.statista.com/statistics/265239/global-oil-consum...>

<https://ourworldindata.org/nuclear-weapons>

<https://en.wikipedia.org/wiki/Nuclear_weapon_yield>

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Notes:

1. 1 GW plant at 17.43 hours ~= 15 kiloton TNT.