I understand your intent, but it should be pointed out that you just described the current federal background system.
The NRA effectively defeats many proposals because the proposals are just added to the current system, and there are thousands and thousands of examples of how the current system has many holes and cracks in it specifically because it is so large and unwieldily.
For example, the NRA points out, using the feds own data, that the smaller, more pointed background systems of the states used when issuing CCPs essentially eliminate any mentally-ill and unstable people from obtaining guns. So, instead of some stupidly large program with many people, let the states handle the process and take this responsibility.
Why the states? Simple - they have already proved that they can eliminate the mentally-ill and unstable from having guns, while the feds have shown they cannot. No need to reinvent a wheel that works with a larger unproved wheel.
Again, your intent is in the correct place and sounds good.
But, no one has addressed the question of why education and training intensive states, such as New York, have the the same accident rate, as states, such as Wyoming, which have the least education and training standards. Until those who advocate expensive federally sanctioned standardized education and training programs, they need to address why New York et al programs do not work in changing the accident rate, if education and training are the keys.
And this is the problem that policymakers have to deal with - when they are presented this data from actuaries using the states own data, it gets extremely difficult to approve funding for a large program which has not proved to be effective anywhere else.
This is similar to the issue with red light cameras. Sounds logical that they would reduce accident rates, but I am part of an insurance company and the data shows that they are not effective overall and in many cases causes more accidents and increases the rate of serious injury accidents. When additional municipal revenue is not the issue, more localities are taking them out, than putting them in.