Do politically-unaccountable bureaucrats in the DOJ (I'm looking at you, Rosey) have the power unilaterally to take someone off the street and vest that person with the power to roam the country and with the same power as a US Attorney indict people for crimes in federal court?
They do it all the time. There are a bunch of DOJ appointments that require Senate confirmation, including the U.S. Attorneys for the 93 district courts.
There is no constitutional requirement for every Federal case to be prosecuted by someone who was confirmed by the U.S. Senate. Most Federal cases are prosecuted by prosecutors who have never been confirmed by the Senate. In fact, constitutional arguments related to this issue can be challenging because the U.S. Department of Justice didn't even exist until after the Civil War.
Here's the text of the U.S. Constitution as it applies to presidential nominations and Senate confirmation (note the bold text):
... and [the President] shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
The bold text clearly gives Congress the power to vest the President or even his cabinet appointees with the power to appoint "inferior Officers" as they see fit. Since the DOJ didn't exist until 1870, the legal authority of the various appointments in the DOJ would have to be determined by reading the various statutes where each position in the DOJ was legislated into existence.
I have seen no references to any statute that supports the contention that the appointment of a "special counsel" is illegal.
I don't think there's any legal basis to the argument that the Office of the Special Counsel is illegitimate. There are, however, potential legal arguments about the scope and authority of a special counsel in a particular case. The Russian defendants have a much better argument to make here than the domestic (American) ones do.
When someone such as an Assistant US Attorney prosecutes a case, they are derivatively using the prosecutorial power of the US Attorney, or in the case of main justice lawyers, of the AG or other principal officer confirmed by the senate.
I don't think there's any legal basis to the argument that the Office of the Special Counsel is illegitimate. There are, however, potential legal arguments about the scope and authority of a special counsel in a particular case.
It is my understanding that DOJ rules predicate the appointment of a Special Counsel on a finding of cause to believe that a crime has been committed, and restrict the Special Counsel's jurisdiction to that crime, and, in the discretion of the appointing officer (AG or deputy), matters arising out of, or related to, that crime, or the investigation into that crime.
In the absence of those boundaries, you do not have a Special Counsel, but an Inquisitor Plenipotentiary.