That is already the law in some places. For example, if the felony conviction was over 10 years ago, and was not a crime of violence or involving firearms, some states allow convicts to petition for a firearms license (at least for possession, if not carrying). Other states require a pardon from the governor, to rescind the conviction (e.g., Delaware requires executive pardon or clemency)
For instance, in Arizona (13-906), a convict can apply for restoration of civil rights, including gun rights, after two years (most felonies), after ten years (serious offenses) and NEVER for any "dangerous offense" as defined. Under Indiana laws, a person convicted of domestic violence crimes can petition for a gun license 5 years after the conviction. Michigan allows petition after 5 years.
In Minnesota you get your civil rights restored automatically upon release from prison, unless it was a violent crime, in which case you need to petition the court for such rights to be restored.
And the list goes on...