Your sentiment is shared by millions of other Americans who fail to see NATO's value, specifically how the United States stands to gain by protecting (objectively) wealthy countries from foreign aggression. The reason we place troops in Europe is entirely political: it's to show our NATO allies that we are committed to the alliance since we hold the biggest military in the alliance (and the world). Additionally, the US places its troops "deep in NATO territory" for military reasons, which as you noted, include increased rapid response capability and projecting military strength.
Also, for perspective, the US military has only placed around 6% of its total troop numbers in Europe (roughly 80,000 out of 1.4 million active-duty soldiers). That means 94% of the US military isn't even in Europe! Sending the troops in Europe home would barely put a dent in our massive military budget, which is now $778 billion.