A common misconception, but incorrect [1]. The Federal Reserve was created by an act of the Congress in 1913 [2]. Part of the confusion arises from private banks holding "shares" in their respective regional reserve banks. These "shares" are less "share of stock" than "you are regulated by us" membership accounts with an archaic label.

The history you mention, of a private organization gradually falling under the purview of public regulation, is closer to that of the Bank of England's [3].

> who lend money to the US treasury at interest paid as dividends

American banks are required to hold a lot of their capital in safe assets, i.e. U.S. Treasuries and things that look like Treasuries. So yes, banks lend money to the government and then get back interest. If you'd like, you can do this too [4].

[1] https://www.frbsf.org/education/publications/doctor-econ/200...

[2] https://en.wikipedia.org/wiki/Federal_Reserve_Act

[3] http://www.intriguing-history.com/bank-of-england-history/

[4] https://www.treasurydirect.gov/indiv/research/indepth/tbills...