It is a given - an institution must be accredited by an ED-Recognized accrediting agency to have some semblance of quality. Well, the expectation is a given; how true that is in actuality is another story. There are unaccredited schools that are strong in education and outcomes while there are accredited schools that are horrible on both counts. No one can argue with THAT.
Similarly, it is a given that accrediting agencies must be recognized by ED to be a good evaluator of institutional or program quality. That is also not true. A very strong case in point is the Accrediting Board for Engineering Technology (ABET) which is THE standard in that particular field. ABET is no longer (by choice) recognized by the US Department of Education. There are other accrediting agencies that are also not recognized by ED - the Association to Advance Collegiate Schools of Business (AACSB) and the National Architectural Accrediting Board, Inc (NAAB), for example. See the full list here: https://ope.ed.gov/dapip/#/agency-list
The basis for ED recognition must be tied to a "federal link" - see CFR 602.10. That is, the connection between the agency and federal funding - HEA programs or non-HEA programs. Yes, the fundamental driver behind ED recognition is so that the institutions or programs can access federal money. Outside of THAT, a link cannot be established and the agency couldn't seek recognition.
So, back to the original question - What is ED recognition really about, really? Is it about quality assessment which is tied up in regulations that are not consistently applied across all agencies or more about making sure that ACCESS to its funds are better "managed"?