I actually know a fair bit about the history of the Federal Reserve and wrote part of my master's thesis on it.

When the Fed was proposed, the bankers were not all that keen on it (despite what you hear on youtube conspiracy videos). They thought they were doing just fine without it, and feared government interference in their work. To persuade bankers to participate in the Federal Reserve system - and participation remains voluntary - the bill promised to put bankers on the governing Board, with each of the twelve member Federal Reserve banks also having bankers on their respective boards. Originally all the board members were to be bankers, both because it was to regulate their deposits and because they were the most knowledgable about their business. Various populist movements at the time campaigned against several versions of the Federal Reserve Act - I recall three, but I think there were maybe four? - and at the end of the day their demand for the governing Board being fully bankers got watered down. Populists in the vein of William Jennings Bryan (who was involved in the bill's creation) were worried that boards comprised solely of bankers would act in their own self-interest, at the expense of rural farmers (which was the reason Bryan gave his cross of gold speech in the first place, and why his hard-sought endorsement of the Federal Reserve Act mattered so much at the time). Instead, the regional banks got a semi-high % of bankers (on the logic that they'd know their regions better than DC / NYC), while the Federal Reserve Board got purely political appointees. Incidentally, bankers hated this and actually fought against it for a long time until it was more or less shoved down their throats.

Presidents traditionally appoint bankers to the Reserve Board on the logic that bankers know their business best, but that's a tradition and not enshrined in law.

In short, the reason bankers are on the Board of Governors is a remnant of the political compromise which was required to pass the 1913 Federal Reserve Act. And to be fair, there is some logic in having board members be bankers, because they do know their business better than somebody who has never worked in it. Obviously it would be bad if all of them were bankers, or if they acted purely in their own self-interest, but that's why there's twelve regional banks providing input on what rates and policies should be, and that's why the Board of Governors is comprised of political appointees and not required to be bankers.

Anyway, back to your Sanders'ing. The history of central banking is a pet topic of mine and I like explaining it whenever possible. Sorry if I bored anyone.