Why should the US punish Ecuador? When did the US become the worldwide punishment police?

Saying "I wouldn't do anything" isn't an answer. You are running a country and you want a "crimminal" returned to the US for prosecution. This isn't the first time in history that this has happened. And the US isn't the only country that does stuff like this. If this happened in China, the UK, Russia, France, etc., I think they'd want their guy back too.

From the WSJ:

Mr. Obama said he hasn't called Russian President Vladimir Putin or Chinese President Xi Jinping concerning Mr. Snowden, who has been charged by U.S. authorities with stealing and passing on government secrets.

President Barack Obama dismissed the idea of using the U.S. military to pursue NSA leaker Edward Snowden.

"I shouldn't have to," Mr. Obama said, addressing a question raised during a news conference with Senegalese President Macky Sall. "This is something that is routine."

Well, you asked what they should do. They should make their case diplomatically and non-threateningly, and hope that they can persuade Ecuador to agree. If they can't, they should respect that this "criminal" is outside of their jurisdiction and there's nothing they can do. They should not attempt to leverage their size and muscle to force compliance, like a bully or tyrant.

Of course, as you noted, it never works like that, here or anywhere else.

And economic treaties are diplomatic tools. They always have been part-and-parcel of diplomacy and it's silly to act like that has suddenly changed in 2013.

Right. I interpreted the original question as "what, ideally, would be the right thing to do?" as opposed to "what would you expect them to do?"

It's not really about the mere involvement of an economic treaty, that can certainly be a part of the negotiations. It's just that, in this particular situation, it's a blatant attempt to use our economic muscle to bend a smaller entity to our will on a completely unrelated matter. (Again, I don't find it surprising or unexpected in any way)

Well I'd argue the minimum proportional response to achieve the ends (i.e. there's some carrot/stick combinations that would not be 'right', or worth the international outcry... and even in that window why go for overkill?)

The problem is that I don't think the U.S. has a lot of carrots at any given time (what are we going to do, lodge a formal protest?), and you'd certainly not want to start off with the stick.

Because we've set a precedent for meddling/policing for over a century, and to stop now would be certain death to the current political party via a media assault.

Our two party (err, one party) system is set up to preserve the status quo in its entirety. To not do so is certain death, politically.

While I agree with you that the US government has overstepped its bounds for too long and they should not be policing anything but their own arrogance, it is inevitable for our government to have to do some sort of ridiculous posturing over this.

I don't think you understand anything about history if you think the US has set any sort of precedent for "meddling/policing" for over a century.