Because they don't trust ANYBODY, that's why: Double Think v. Cognitive Dissonance

Newtown group seeks city-wide ban on carrying guns in businesses:
February 07, 2016

The Newtown Action Alliance, a gun control group based in Connecticut, has sent a letter to all of their local businesses requesting that they make their properties gun free zones, and not allow firearms to be carried on their premises. 

As is typical of all gun-control activists, this group does not believe that anyone can possibly justify being armed in public places.

Why? Because guns kill people.

These people would rather depend on their local sheriff to defend the public than anyone with a concealed carry license.  Because they trust the sheriff, but they don't trust armed civilians.

The problem is, these Newtown folks (and others with the same mindset) find themselves unable or unwilling to accept the "vetting" process by which their local sheriff determines that specific Concealed Carry License holders are legally determined to be a minimal threat to the community.

Which is to say, they trust their local sheriff to rescue them when they are threatened, but they do not trust their local sheriff to reliably identify and approve non-LEOs to possess the means to defend themselves (and others) when 'the police are just minutes away'.

There are two psychological term for this: the first is Cognitive Dissonance

In psychology, cognitive dissonance is the mental stress or discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values at the same time, performs an action that is contradictory to one or more beliefs, ideas or values, or is confronted by new information that conflicts with existing beliefs, ideas, or values.
Leon Festinger's theory of cognitive dissonance focuses on how humans strive for internal consistency. An individual who experiences inconsistency (dissonance) tends to become psychologically uncomfortable, and is motivated to try to reduce this dissonance—as well as actively avoid situations and information likely to increase it.

The other psychological term is known as Double Think, which is the exact opposite:

Doublethink is the act of ordinary people simultaneously accepting two mutually contradictory beliefs as correct, often in distinct social contexts. Doublethink is related to, but differs from, hypocrisy and neutrality. Somewhat related but almost the opposite is cognitive dissonance, where contradictory beliefs cause conflict in one's mind. Doublethink is notable due to a lack of cognitive dissonance — thus the person is completely unaware of any conflict or contradiction.

{emphasis added}

Which to say, Cognitive Dissonance makes the stress uncomfortable, because the suffer just doesn't know WHAT to think.

Doublethink, on the other hand causes no distress to the sufferer.  They know that they accept contradictory beliefs, but they don't consider that a personal problem.  (As it ... "it's not me, it's THEM!")

which leads to this:
Newtown Action Alliance