I do agree that the central bank is essentially a bit of the private sector that acts as government. It is the apex of the private banking system, a private enterprise (though it does by law remit any profits it makes to the central government) that does make decisions that affect the whole nation.

There is, however another option, using a “democratically distributed income” to supply the economy with money (as currency). The amount of the supply of money would be determined by demographics — and only that. It would provide the means to making the existing economy wholly self-regulating (with built-in safeguards against inflation) while eliminating unemployment, poverty, taxes, and public debt and increasing sustainability. If curious, a place to start would be “A Truly Great Idea” published in Data Driven Investor in Medium.