In the Cold War, as you allude to, there were enemy submarines with intercontinental nuclear missiles waiting hidden in the depths, and they would have to be destroyed in order to prevent the missiles from being unleashed upon cities or military bases. That’s not the case in the situation where the US faces off against any of its current potential adversaries with submarines (Iran, North Korea.)
Since the 1960s, the submarine, which generally had the role of destroying enemy warships or merchant shipping, have had bifurcated roles:
(1) there are attack submarines which still have the old role - the destruction of enemy warships and merchant shipping
(2) there are missile submarines which have a strategic role - the last resort hidden arsenal of long-range nuclear weapons that can be used strategically against enemy installations and homeland.
While both North Korea and Iran have submarines, the only submarines they have are the 1st type - attack submarines. Because of this, it’s unlikely that a war would involve finding submarines with the primary mission of staying hidden before they launch long-range missiles. The Iranians and North Koreans don’t have submarines that can launch such long-range missiles, and they don’t have a navy that could pose a threat to US missile submarines that would be stationed too far away from Iran or North Korea.
It’s much more likely that the US Navy would be responsible for the protection of civilian and military shipping in the waters near Iran or North Korea in case of such a war, and there we would see attempts by Iranian or North Korean submarines to penetrate and attack convoys or warships, and a sophisticated US Navy put up against lower-tech submarines. According to publicly-available information (like the Harpoon naval games,) the US edge in antisubmarine warfare would be a huge advantage, but who knows how that would play out in reality.
It’s conceivable that the US could one day go to war with one of the nations that does have long-range missile submarines, like China or Russia, or much[sup]umpteen[/sup] less likely the UK or France. If so, the US would not enjoy the technological antisubmarine edge it enjoys over Iran or North Korea. Who knows how that would turn out - guessing what would happen in a war before it happens is risky.
Your question is kind of interesting because submarine warfare has really always been a form of asymmetrical warfare.
The large, wealthy naval power (the UK, US) goes in for a surface fleet that can’t be matched by its adversary, so the adversary (Germany, the CSA) goes for a less expensive submarine fleet in an attempt to defeat the wealthy naval power without attempting to directly confront the powerful surface fleet.