http://www.pewresearch.org/fact-tank/2013/06/04/a-minority-o...
Arresting somebody who owns one or two guns is really not inherently safer than arresting somebody who owns a few crates of guns, so the number of guns in the country is irrelevant. The number of gun owners is relevant.
The apparent upward trend in police militarization and the number of Americans that own guns simply do not correlate. The Economist's unfamiliarity with American culture (particularly American gun culture) explains how they could make this mistake.
Police in America are less likely today to encounter guns than they were 40 years ago, but police shootings are apparently at all time highs. If their theory held water, that would not be the case.
Your hypothesis does not fit the data.
They qualify the above statement in the prior paragraph: "One reason why so many American police shoot first is that so many American civilians are armed."