(Just one unpopular opinion; I've seen some very good arguments on the other side here as well.)
I don't have any particular heartburn either with a background check, and/or required training / certification, nor with denial to felons.
The problem is that some of us would like to lump all 'rights' into the same box of risk / reward to the others around us. Common sense says it doesn't wash.
While true that speech, religion, the press, and other rights can be and are misused, none of them are immediately fatal in the same way guns are, and none of them are capable the instant and irreversible damage guns can cause. With those others there is normally a chance to intervene, and time to respond. It is fact, and clear to everyone, that guns are inherently immediately dangerous in ways those other rights are not.
Other people aren't stupid, they know this. It's foolish and insulting to try persueding them that this difference doesn't exist, and to me such arguments make us look either stupid, or fanatical, or both.
It follows then, at least to me, that it's reasonable for them to require those who choose to exercise the more dangerous right be required to demonstrate at least minimal competence and a history of mental and social responsibility before allowing it, and we should be helping, not hindering. It may be annoying at worst, but there are plenty of ways to do it and still prevent abuse. As such it is not an infringement.
Felons, well, they made their own choices. It's a sadness that some of these people do change and would be fine with a gun, and it's a fact that too many do not and should never have one. And often impossible to tell the difference, and I'm not sure there is any particular benefit in forcing the rest of us to try. The costs of felony crimes are no secret, and anyone who neglects to calculate in advance should be prepared to pay them. Can it be made perfectly fair? No. Nor can anything else.