Wasn't that the legal genius in how Trump structured it? DOGE is a re-branded USDS (United States Digital Service).
Given that USDS is not a federal agency, that clause of the Constitution would not apply.
That also would assume that Musk is the administrator of DOGE, which he is not. It's Amy Gleason, and she, just like every administrator of the USDS going back to Obama, was not confirmed.