I do not think it is such a problem.
I am just not sure they are not vetted.
I googled it and I did find that some of the DOGE employees are holdovers from the US Digital Services, because
NPR told me. I would imagine that employees from US Digital Services were vetted. I am not sure that the folks who came in since the inauguration with Musk are not vetted in any way.
And I am not sure what kind of vetting government employees who need access to this information to do their jobs (say, a Social Security Administration claims rep) actually go through. If you are a teacher, I'd bet you went through a more rigorous vetting process. For those who will be looking at secret information, those will need a secret security clearance, but issuing an interim clearance gets done pretty quickly. For higher level clearances, the process is more in-depth.
At any rate, I suspect that politicians and partisan media stating (or implying through clever phrasing) scary things like "All DOGE employees, none of whom have undergone any vetting process, have access to all your most sensitive personnel information and unfettered access to the government's most secret information," is just not true.