"Until this week, government officials had resisted answering inquiries as to who was formally in charge of [DOGE], except to say that it was not Mr. Musk."

"(Nor is Mr. Musk among its employees, the government said.) On Tuesday, a White House official said that Amy Gleason, a former health care investment executive, was serving as the acting administrator. On Friday, Joshua E. Gardner, a lawyer in the Justice Department’s civil division, denied that Mr. Musk had any role with the Department of Government Efficiency. This despite Mr. Musk’s clearly driving its initiatives, including an email blasted out last weekend that attempted to require all federal employees to respond with a list of five accomplishments from the previous week. Although the email was sent by the Office of Personnel Management, the federal government’s human resources arm, Mr. Musk said on Wednesday that he had suggested it and that the president had approved...."

"Judge Appears Skeptical of Claims That Musk Isn’t Driving DOGE/The judge prodded government lawyers for additional clarity on Elon Musk’s role in a case that directly challenges the constitutionality of his operation and his part in the rapid reshaping of government" (NYT).

The argument is that the Appointments Clause applies and Senate confirmation is required because Musk is a "principal officer" and not an "inferior officer." I'll just give you this quote from the 1988 Supreme Court case, Morrison v. Olson:

The line between "inferior" and "principal" officers is one that is far from clear, and the Framers provided little guidance into where it should be drawn. See, e.g., 2 J. Story, Commentaries on the Constitution § 1536, pp. 397-398 (3d ed. 1858) ("In the practical course of the government, there does not seem to have been any exact line drawn, who are and who are not to be deemed inferior officers, in the sense of the constitution, whose appointment does not necessarily require the concurrence of the senate"). We need not attempt here to decide exactly where the line falls between the two types of officers, because, in our view, appellant clearly falls on the "inferior officer" side of that line. Several factors lead to this conclusion.

First, appellant is subject to removal by a higher Executive Branch official.... Second, appellant is empowered by the Act to perform only certain, limited duties... Third, appellant's office is limited in jurisdiction.... In our view, these factors relating to the "ideas of tenure, duration . . . and duties" of the independent counsel... are sufficient to establish that appellant is an "inferior" officer in the constitutional sense....