A possession and acquisition license already is a privilege not a right, and the application includes a series of questions (some as invasive and insulting as they are absurd) that concern that onus you are talking about. I expect many here would scream bloody murder if they were asked of any demographic other than people who use firearms.
Thing is, law doesn't mean a damned thing without the means and the will to enforce it. It's no different with firearms than it is with booze, drugs, taxation, non-discrimination or anything else.
On the other side of the equation, I expect that for some no burden of proof would be sufficient to allow firearms use. There are certainly those who make that argument.
So I am not sure "we" have any solid consensus here.