Instead of appointing new cabinet officials to monitor the phase out, why not just not avoid appointing anyone? Accept resignations, start transferring people and hiring freeze on all department hiring and zero budgets.
It is nowhere that easy. Federal agencies are somewhat like malignant tumors, sending tendrils out, in the form of existing contracts, regulations, pending lawsuits, funds to disperse, etc., etc. And GS personnel are protected from being dismissed by the government employee union.
As a wild guess, to shut down an agency may cost between $500 million and a billion dollars.
And if the agency does anything that congress still wants done, they must legally transfer that function to a different agency.
Add it all up, and you need someone at the helm to guide that agency to the scrap heap, and they must have the full backing of the POTUS, the congress, and no interference from the courts to do it as well.