Having elected officials is not what constitutes a public institution. CFR 417.201 defines a public institution as “an institution that is operated by or contr

Having elected officials is not what constitutes a public institution. CFR 417.201 defines a public institution as “an institution that is operated by or controlled by the Federal government, a State, or a political subdivision of a State such as a city or county.” For example, as you stated the government sets the Fed charter and thus enumerates their powers and duties. Congress also pays their salaries, so (to a very limited extent) the Fed receives government funding to operate and reports to Congress for accountability. Most federal “public institutions” do not have elected officials, but rather are led by appointees, just like the Fed.