From Who owns the Federal Reserve?
The Federal Reserve System is not "owned" by anyone.
Some observers mistakenly consider the Federal Reserve to be a private entity because the Reserve Banks are organized similarly to private corporations. For instance, each of the 12 Reserve Banks operates within its own particular geographic area, or District, of the United States, and each is separately incorporated and has its own board of directors. Commercial banks that are members of the Federal Reserve System hold stock in their District's Reserve Bank. However, owning Reserve Bank stock is quite different from owning stock in a private company. The Reserve Banks are not operated for profit, and ownership of a certain amount of stock is, by law, a condition of membership in the System. In fact, the Reserve Banks are required by law to transfer net earnings to the U.S. Treasury, after providing for all necessary expenses of the Reserve Banks, legally required dividend payments, and maintaining a limited balance in a surplus fund.
The wikipedia article from which you quoted states (just a couple of sentences before the quote in the OP)
The Federal Reserve Banks have an intermediate legal status, with some features of private corporations and some features of public federal agencies.
Public-private cooperations work. Why break something that not only isn't broke but that works? This intermediate status between public and private has proven useful in a number of other areas.For example,
The Jet Propulsion Laboratory (JPL), which has recently landed yet another rover on Mars (and has recently flown the first helicopter on Mars), is a Federally Funded Research and Development Center (FFRDC). Its employees are not civil servants. They instead are employees of the California Institute of Technology. There are 41 other FFRDCs. This list includes almost all of the institutions that worked on the Manhattan Project.
The Applied Physics Laboratory (APL), which somewhat recently flew the New Horizons spacecraft by Pluto, is a University-Affiliated Research Center (UARC). There are 13 other UARCs that perform research at the behest of the federal government. The UARCs are a bit more free than FFRDCs with regard to performing research for non-governmental entities. Like the FFRDCs, the UARCs are required to be not-for-profit organizations.
The Federal Reserve Banks, the JPL, and the APL are but three of many examples where public-private organizations have played a key role in the last century. Yet another example: without public-private cooperation, not just in the US, but also globally, we would not have had vaccines for COVID-19 nearly as quickly as we did.
Entities that are on the border of public and private work. They work quite well. If it ain't broke, don't fix it.