I'm not necessarily advocating this but I think the following would address most people's issues, with the exception of those people who want no restrictions of any kind.
Three classes of firearms requiring 3 levels of training,commitment,background checks.
Class 1 - rifles, shotguns, and handguns that do not have removable magazines and can hold a max of 6-7 rounds. So revolvers, bolt-action rifles, most shotguns.
Class 2 - semi-auto rifles and pistols that fire one round per human finger pull. magazines limited to 30 rounds.
Class 3 - whatever weapons are currently in the category of automatics, and devices that enable class 2 weapons to fire without a human trigger pull.
Class 1 would require about the same thing as is currently required except background checks for all sales and limits on the number purchased which would probably require some sort of record keeping.
Class 2 additionally would require registration and some sort of renewable license that would include ongoing background checks and training. But Class 2 should not be as restrictive as class 3. The goal wouldn't be to restrict ownership but to require a higher level of commitment.
Class 3 restriction would remain the same.