If we can’t have independent Executive Branch agencies, then Congress is probably going to have to create some under Congress. They do have some bureaucracy of their own, like the Congressional Research Service.

https://en.wikipedia.org/wiki/Congressional_Research_Service

The Congressional Research Service (CRS) is a public policy research institute of the United States Congress. Operating within the Library of Congress, it works primarily and directly for members of Congress and their committees and staff on a confidential, nonpartisan basis. CRS is sometimes known as Congress’ think tank due to its broad mandate of providing research and analysis on all matters relevant to national policymaking.[4]

CRS is one of three major legislative agencies that support Congress, along with the Congressional Budget Office (which provides Congress with budget-related information, reports on fiscal, budgetary, and programmatic issues, and analyses of budget policy options, costs, and effects) and the Government Accountability Office (which assists Congress in reviewing and monitoring the activities of government by conducting independent audits, investigations, and evaluations of federal programs). Collectively, the three agencies employ more than 4,000 people.[5]

The whole federal government is about three million people, so that’s a tiny fraction of it. But it is a thing – the Legislative Branch can have its own bureaucracy.

I mean, you cannot stick stuff related to oversight of the Executive Branch within the Executive Branch if it doesn’t have independence.