Part of the resistance towards requiring training is that it provides a means to abridge the 2nd Amendment. It's possible to make it very difficult to acquire a gun by making it very difficult to get the training necessary. Perhaps you only offer it in one location, perhaps you make it expensive, perhaps you only offer the class once a month and have a class size of 20 people. It's how California has basically banned concealed carry.

I think it would make more sense to just make the training universal, integrate it into phys ed in high school or something like that. I don't know that I would integrate actually firing a handgun, but some general information would be good I think. Things like where the bullets would come out, how to check that the magazine is empty/ejected and the chamber is clear, how to safely clean a gun, the kind of things that bullets will penetrate (I hear people thinking that stuff like 2x4s or drywall will stop a bullet sometimes). Just general information that you should know any time you will be handling or near guns. The gun ownership rate in this country is very, very high, I don't think it's imprudent to make sure that the populace knows at least enough to unload a gun and put the safety on.