The US Cares? You mean like career ivy league politicians? The average american gives zero fucks and just wants healthcare and jobs and shit. Oh and to not die in a war halfway around the world.
This nebulous idea of "influence" bothers me lol. Lets spend billions and billions and billions of dollars for this fake currency monopoly money called influence. I want to see the receipts. What do we get out of "influence" and is it worth what we are spending?
OK, I'm going to give you three alternative realities to demonstrate why influence over Europe is of enormous importance to the shape of the US itself.
1) Imagine a world where the US immediately retreated from Europe after WWII and turned isolationist, as it had been before the war. NATO never forms. In 1949, the USSR gets the Bomb. Stalin gives Europe an ultimatum: accept the socialist revolution or perish in nuclear flames. Europe submits. The USSR now stretches from Vladivostok in the east to the English Channel in the west: the greatest empire the world has ever known. The US is cut off from trade with Europe. The US economy collapses. The USSR emerges as the one world superpower, while the US is relegated to a second tier power similar to the Russia of today, with a similar disparity in living standards.
2) Imagine a world where neither the US or the USSR intervenes in Europe after WWII for whatever reason. Within twenty years - just like after WWI - the nations of Europe are at each other's throats again. Except this time, with nukes. Europe is immolated. The world is thrown into a nuclear winter. Half the US population starves to death, the other half envy the dead.
3) Imagine the US electing a Trump-like figure instead of Bill Clinton to replace Bush Sr. The US leaves NATO, citing rising costs and no purpose, and pulls out of Europe. It bullies the EU over trade. US-EU relations break down and never recover. Over the next 40 years, the EU grows ever closer and finally federalises, with a combined economic and military strength that's more than a match even for the US. Except this EU is about as friendly to the US as present day China is and mercilessly competes with the US over world trade and dominance, even going as far as militarising Greenland and forging alliances in South America. The US enters the 2030s with not one rival super power in the shape of China, but two, and with greatly diminished soft power and trade to boot.
Sure. Call this Clancy stuff all you wish and that it's just fantasies. But I can assure you, the US has maintained influence in Europe all these years precisely in order to avoid outcomes like this.