You are right to be suspicious of plausible sounding hypotheses that aren't backed by empirical evidence, but you are failing to correctly account for prior probabilities. The prior probability that any particular phenomenon
is causally linked to a given phenomenon is very low (and as a corollary, the probability that one phenomenon
is not linked to another is very high).
Unfortunately, humans are pretty bad at handling Bayesian inference implicitly, so we have to use some tricks to keep ourselves in line. A particular problem that we face is that once we start investigating a hypothesis, we seem to upgrade our prior for that hypothesis, and so when we find weak evidence, we become overconfident in the hypothesis. This is related to the prosecutor's fallacy[1], and is also known in rationality circles as "privileging the hypothesis"[2].
The other bad thing about privileging the hypothesis is that we only have so much time to investigate hypotheses, and we have a ton of bad hypotheses. In fact, because of Occam's razor, we actually have way, way more bad hypotheses than we have decent ones.
So investigating hypotheses with low priors has a low ROI
So in summary, we don't really need to (and shouldn't) gather evidence to see if video games can be linked to mass shootings because the hypothesis "video games aren't linked to mass shootings" has such a massive head start over "video games are linked to mass shootings". If, however, simply surveying the data available at the outset revealed significant reason to believe that video games might be linked, it would be different. That's why it is worth investigating the link to guns, even though the prior for that connection was also low.
[1] http://en.wikipedia.org/wiki/Prosecutor%27s_fallacy
[2] http://lesswrong.com/lw/19m/privileging_the_hypothesis/