I don't think that the driver's license analogy is a good one, because like you said I too believe that owning a gun for self defense ought to be considered a right, whereas driving on public roads is considered a privilege.
My problem would be if the State required State administered safety tests. Then you have a collection of government pukes deciding if you are "safe" enough to be able to defend yourself.
Of course, they might decide that guns are inherently unsafe, and an affront to public safety, in which case, the bureaucracy will decide to "save" you by absolutely refusing to allow you to handle a firearm. Or even something a lot more pedestrian, but still terribly insidious, such as the clerk just doesn't believe in guns ownership and drags his feet, or somehow subverts the process every chance he gets.
The right to keep and bear arms is too important a civil right to put into the hands of bureaucrats in that way.
Having said that States that allow CCW/CHL are obviously requiring a competency test before issuing the permit. If this is to be done, then it ought to be allowable, as it is currently, to have private/third party entities administer the class and test so as to keep the influence of government away from the process as much as possible.
Should you then say, well let's make the test harder to pass, well then two problems would arise: 1) you might possibly infringe upon a person's right of self defense, that can shoot safely, but perhaps not at the level of a grade A pistolero, which wouldn't be right, and 2) Anti Self Defense types would charge that it isn't right that regular people should be forced by the State to learn "combat" training designed to kill people.
So, as it is, the status quo is probably the best compromise, that is, yes have a basic course and a basic competency exam for CCW/CHL carry only, administered by third parties, to make sure that you can handle the weapon safely, but to also allow the citizen to get more advanced training on his/her own.
After all learning how to use a gun safely isn't rocket science. And forcing people to have to learn advanced shooting techniques just to allow them to have access to guns or CC, would be akin to forcing an average driver to learn advanced stunt/race car driving before you give them a license.
I'm all for getting as much shooting (and driving, for that matter) training as possible. But this should be an individual's choice, not a matter of governmental coercion.
Anyway, just some initial thoughts that I had.