Central banks are the government. And governments define the law we all operate in.
And as the crypto industry experiences today they can be vindictive, inconsistent, spurious, aggressive etc. And are highly skilled at resisting efforts to limit their power.
The Federal Reserve Bank of the United States is, despite the misleading name, a private corporation. [1] This is true I believe of many central banks. Nevertheless, and perhaps this is what you really meant, central banks can _act as an instrument of government_, or, more cynically, serve to enrich the political class and politically well-connected.
That link does not at all describe "a private corporation" in the sense that people who don't click the link would assume your use of the phrase would imply.
A better summary of that link would be "it is a legally murky thingamajig, which is symbolically private while being governmental in practice".
The section concludes with this quote from a political science professor:
> the "ownership" of the Reserve Banks by the commercial banks is symbolic; they do not exercise the proprietary control associated with the concept of ownership nor share, beyond the statutory dividend, in Reserve Bank "profits." ... Bank ownership and election at the base are therefore devoid of substantive significance, despite the superficial appearance of private bank control that the formal arrangement creates.
Now, it's very possible that the Wikipedia article is itself misleading. I'm just saying it isn't a good citation to use if what you want to do is casually describe the Federal Reserve as "despite the misleading name, a private corporation". The section you linked is all about how that would be a more misleading description!
And as the crypto industry experiences today they can be vindictive, inconsistent, spurious, aggressive etc. And are highly skilled at resisting efforts to limit their power.