Initially, America very much wanted it for strategic reasons. The GIUK Gap was hugely important. Specifically, it was important to the Soviet Union's submarine fleet.

If you look at the terms of the Montreux Convention, it was impossible to "sneak" a submarine through Turkish waters. If you look at a map of the Baltic Sea, or more specifically the waters around Denmark, it's similarly unlikely that you could ever sneak a submarine past even a semi-aware detection network. Denmark, of course, was one of the founding dozen of NATO.

This means that if the Soviets actually want to conduct any submarine operations with any degree of stealth, they need to be based out of Murmansk in the White Sea or somewhere in East Asia (Vladivostok, Petropavlovsk on Kamchatka, Magadan, or Sovetskaya Gavan. Realistically, if the Soviets wanted to have their submarines remain undetected, they really could only use those five ports. And if you wanted to operate in the Atlantic, you weren't going to put your HQ on the northwestern coast of the Pacific. You were going to put it in Murmansk.

This, effectively, meant that any ships the Soviets sent to the Atlantic had to pass through the GIUK Gap. And it would be relatively easy to detect (and subsequently track or shadow) them if you had assets in the area beforehand. And it's kind of hard to be all sneaky and such when the USN is dropping practice depth charges on you. They couldn't do any meaningful damage, of course, but it's an implicit threat: the USN was basically saying, "we could sink you at any time."

As for the overall economic value of Greenland: that's up in the air. We already see Greenland being exploited for hydrocarbons. The USGS released a review of hydrocarbons in the region here. I believe they described it as "a genuinely stupid amount of dead dead plants buried in the sea floor." (Okay, that wasn't their exact phrasing.) (._.)

Theoretically, yes, Greenland would've paid for itself. Eventually. If the estimates actually pan out.