The Nazis did come to power mostly through success at the ballot box. They had tried to take power by force (see the Beer Hall Putsch) but had failed. However, they had organized gangs of hoods that were intimidating political opponents using violence as a means, with corrupted police turning a blind eye, that helped forment their ballot successes. These groups (that would later become the SA and the SS) would not have had as much success in intimidating political foes to the Nazis were the populice armed, and the author of this piece fails to mention that.
Further, when General Beck and several others in position to attempt a coup against the Nazis were formulating their plans, which they did fairly often but nearly always got cold feet over, they realized that the people could be no help due to brainwashing by propaganda and because they had no arms. The army was the only hope, and they never felt confident enough in their chances for success. If the people were armed, then a counter-propaganda campaign could have been used to rile up some armed resistance to aid in a potential coup.
Gun control was not enacted by the Nazis. But as someone mentioned above, the Nazis took full advantage of the consequences of gun control. It was perhaps the most poignant example yet of the edict that gun control only disarms the honest and leaves the thugs armed.
Hitler came to power through intimidation, murder, and the unfortunate fact that the mainstream German press, and the international press, never picked up on the exposes printed in Munich (for example, the death of Hitler's niece, which was clearly a case of Hitler murdering her--shockingly, with incestuous overtones). Not to mention that Hitler had the south German judiciary on his side.
His rise to power had very little to do with gun control.