Can you prove the opposite has happened because every journalist says otherwise? The very article you are commenting on suggest the FBI did not do background checks because if so this particular person wouldn't have been approved. I'm all for having a committee that scrutinizes line items but the ends don't justify the means. Especially because there is no oversight here.
What evidence do the journalists have? Elon is cleared, many of his employees are by nature of the work they do. Many of the listed personnel for each of the DOGE teams in the orgs are comprised of Cleared lawyers and invdividuals from within the orgs as well. The burden isn't on proving they aren't, it's on the journalist to prove the sensationalist claims. There is clearly evidence of oversight, the president is authorizing actions. He was elected, we don't have to like it but it's how things are structured. Want change, back the candidates that will fix the issues you want, convince everyone else to agree.
One good example of sensationalism from journalists is the claim this is a "Data breach". That's neither true, nor helpful.
That is not how oversight works. Oversight is an unrelated non-partisan committee and transparency. Unfortunately Elon is jumping through hoops to avoid transparency like moving off any communication that would be subject to FOIA. What you describing is a crony doing his masters bidding not oversight and transparency.
You clearly have your perspective and the rest of the population has theirs. If he / they broke the law, I encourage you to engage your representatives and push for the appropriate actions. Take charge. Until then, it is what it is.