big deal–
The high-capacity magazines that Lanza used allowed him to shoot a huge number of bullets before the police arrived at the scene. Each of his victims had at least three gunshot wounds. This type of magazine would be prohibited under the new law. While there still would have been victims even if seven-bullet magazines were used, there probably would have been many less.
What are you unsure of? The other two shootings were committed by known mentally ill individuals who bought their own guns. Background checks, which the new law would mandate together with reporting, should have stopped those sales.
As far as inconvenience, a one-time background check when buying weapons is far less inconvenient than the screening we must undergo when flying, when going to a stadium (e.g. for the siyum hashas), when going into a mall in Israel and many more examples. BTW, you are aware that this inconvenience already exists whenever buying from a gun store?