Carriers would provide air superiority to prevent any reinforcements from landing. That's all. Trump is willing to push a war of attrition on this one, because again, he's an idiot.

If a nuclear sub did manage to sink a US supercarrier, Trump would likely try to galvanize support in a "Remember the Maine!" fashion. I doubt it would shift actual support much at all. Quite the opposite; without doing the legwork for a paper thin excuse ahead of time and building a media frenzy the way the Bush II admin did, it only highlights how dumb and pointless the whole thing is, and that he's putting American service members at risk for no gain whatsoever.

Oh, and that nuclear sub would be hunted down and sunk in response. The EU doesn't have the forces to win a war of attrition.

The only way I see this working out for the EU is if there's a major purge of the US military beforehand. That would ensure loyalty to Trump, but it would toss competence into a woodchipper. Edit: see the Finish Winter War for how well purges work out. And the Finns technically still lost that one.