To be fair, barry never allowed any gun parts (bump stocks) to be banned. He did try 2x and failed.
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The only thing Obama did in the 2A arena was:
1. Create more demand for guns and ammo (and thus more production, imports and sales) than any other president in 40 years, and
2. Appoint VP Joe "Do Nothing" Biden to head up hearings to gather ideas and push for more gun control laws. BTW, this was in the wake of Sandy Hook, when he thought the emotional momentum that event caused would overpower any logic and reasoning 2A supporters could muster.
The main reason nothing happened to advance gun control is because Obama and Biden were 100% incompetent and ineffective. After Benghazi and Obama all but admitting the Obamacare sales pitch was a pack of lies, nobody paying attention trusted anything he had to say.
Obama laments that he wished he could have done more to enact "common sense" gun laws, so it's not from lack of desire.