The government background check for buying firearms does not involve fingerprints either.

It is required for security clearances. For those positions you also give up 4A protection.

I attest to this. I have fingerprints filed with at least 10 different civilian fed agencies.

Depends on which firearms. NFA items absolutely do need fingerprints.

True but that's less than 1% of firearm purchases. Normal 4473 purchases don't require fingerprints or photos or anything like that.