Even traditional Medicare is mostly administered by private companies, including all of the large for-profit health insurance companies. Medicare for All will not necessarily change this - the government may still choose to contract with private companies for much of the administrative work. But there will be less of this administrative work with a decently designed publicly financed system than with the fractured, multiple risk pool system we have now. See Private sector vs. Medicare? It's basically the same thing .
While it is not “basically the same thing", since it matters who makes the rules, how much profit is allowed for simply bearing risk, and how many different risk pools are allowed (one is the number improved Medicare for All advocates want), the article is correct that there may still be plenty of contract work for insurance companies after we implement improved Medicare for All.