Why would government work automatically allow for a background check? If anything, government should be more neutral than private employers who should be allowed to use their own discretion when spending their money. Government procurement is so focused on appearances. I'd definitely support government not discriminating based on crimes for which someone had paid their debt to society.
Many private sector companies handle data that the general public probably shouldn’t. Extending special protections to the state that are not allowed to private employers is ridiculous and unsafe.
Companies like credit bureaus or banks or the telcos or airlines or hospitals are effectively arms of the state at this point. You don’t get to opt out of the oligarchy with consumer choice.