To your earlier point about healthcare falling under the clause of "life, liberty, and the pursuit of happiness" and therefore a right, I would also draw the same analogy to gun falling under the same conditions.  The right to life does include within it the right to not be killed and therefore owning a gun can be a measure of self defense.  Within this argument, the right to own a gun seems to be justified.  We can for sure debate then what types of guns are best suited or not for this, and it would at least seem to me that since a high powered rifle is primarily a "long" distance weapon (as opposed to a hand gun, at least) is not. It's hard to make the argument for self defense with a gun like the AR-15 since your target is normally far enough away from you that he's not posing such an immediate threat that shooting him is your only option.
But I definitely agree with the latter part of your argument.  Owning a highly destructive weapon like a gun needs to require proper training and education, something at least similar to a driver's license, preferably a pilot's license even.  Additionally, not only should background checks and mental health checks be performed before the sale of the gun but they should be done routinely post-sale as well.  The government doesn't need to take away guns, but it does need to ensure the safety of its citizens, which is one of its primary and direct duties.  There can be far more done on this topic long before the conversation of "taking away/banning all guns" comes up.