Three cheers for not overreaching one's mandate. I know we rag a lot on Federal Reserve chairpeople for not doing NGDP level targeting, but in a lot of ways it's one of the most sensible branches of government.
This is an amazingly misleading article. What Yellen said was: the Fed only has the authority to regulate banks, and if Bitcoin stays extra-bank, then it has no authority to regulate it. However, she said, the Department of Treasury <i>does</i> have the authority to regulate it.
It's kind of funny, but on one hand, that statement is probably the best thing a libertarian reader could imagine hearing from Yellen. That's how I read it at first, and was thinking, "wow, that's pretty radical for the government to intentionally take itself out". Yet, upon second reading, I think the intention was probably more to cast Bitcoin as a wild-west, do-at-your-own-risk, good-luck-to-ya type of asset to work with. I think the message was meant more for banks and investors. The real subtext being, "if you lose your shirt, get defrauded, or there's large scale tampering, don't come crying to the Fed".
Fair warning: she means the Federal Reserve doesn't have the authority to regulate Bitcoin. The Federal Reserve is not the nexus of financial regulation in the US Government, and Congress can delegate additional authority to the Fed any time it wants.
The letter Manchin wrote was not to the Fed but to the secretary of the Treasury and a few other Treasury-related officials. They haven't responded, as far as I know. This is a separate (and very dumb) line he is pursuing. I consider myself fairly ignorant of government issues like this but even I could tell you the Fed would be the wrong institution to ask about regulating Bitcoin (already a foolish notion).