A criminal record is not necessarily a reason to provent someone from owning, possessing, etc. a gun. A crime of violence is; as is a mental condition wherein a person has a demonstrated history of little or no self control, or has a history of substance abuse.
Consider the most recent mass murderers were not criminals until they entered a school or movie theater or political rally with a gun and began shooting. A BGC is much more than simply looking at a history of arrests.
[Before hiring LEO's we would look at the entire record of the candidate, including civil judgments, credit reports, marriage history, residence stability, job stability, school records, military records and FI cards (police field interviews) of police agencies in all city/towns wherein the candidate lived, and review refernces with neighbors, teachers, coaches and family members as well as state and federal arrest/detention records. Plus, a written and two interviews with a psychologist and at least two interviews with command staff.]
So what's the big deal about BGC's for civilians seeking to purchase a gun. If the buyer wants a gun, and has no nefarious plans why wouldn't the buyer be as willing to undergo a BGC? BTW, there is no right to privacy nor is their funding sufficient to carry out an extensive BGC for all civilians who want to buy a gun.
We could start now, and require all new purchases of guns to include a BGC. Thus all the gun lovers can keep their guns safe and secure from government oversight; at least until they violate laws restricting the sale of guns, commit an act of violence, demonstrate a lack of self control, are detained as a danger to themselves or others or have been determined to be a drug or alcohol abuser.