This was a very interesting video. There is actually a lot going on in the background about actual statistics (confidence intervals, significance/p-value, null/alternative hypotheses) and this video sums it up in layman's terms quite well. The whole study of statistics is about proving (or disproving) that correlation equals causation, that is, in this case, a "gun control policy did or did not have the desired outcome as a result of its implementation".
However, people should not use this video to "disprove the effectiveness of gun control". The same statistical problems affect both sides of the argument, namely a poorly constructed hypothesis and the lack of quality data. I also believe the author is correct, that it is certainly possible to use statistics to "suppress" information, and without rigorous supplementary analysis, it is hard to discover.
Think "averages". If you do an arithmetic average (sum all parts then divide by the number of parts) you can hide the fact there is a very large range of data. For example: 1 + 50 + 100 has an 'average' of 50.3, and the 'median' is 50. If I used this method to describe "gun control by year", it would appear that the average of mass shootings or whatever has not changed over 3 years, when in fact there was a dramatic change. If we had used standard deviation and variance, we would see that the data points are an 'average' of 49.5/2450 from the 'mean', indicating a HUGE dispersion in the data. So, if we were given the 'average' of 50, but also the SD and Var of 49.5 and 2450, we would know something is really strange with this data even if we could not see the data itself.
Gun control is going to be a huge debate for a while. I was going to write more, but it's time for lunch!