One basic principle of “getting things done” is that authority and responsible have to be matched. If you make someone responsible for doing something, then you have to give them authority to do it. If you what to give them authority, then you better make them responsible for the consequences.
In the case of currency exchange rates, making Treasury responsible for it is silly if you want anything done, because they have zero authority to do anything that might actually change exchange rates.
Now the Federal Reserve *does* have authority to set exchange rates. The curious thing is that it has no responsibility to do so. Under the Humphrey-Hawkins Act, the responsibility of the Federal Reserve is to maintain growth and minimize inflation. The interesting thing is that currency exchange rates isn’t mentioned, and even more interesting neither is balance of trade.
Part of the reason this is the case, is that under Bretton-Woods the institution that maintained exchange rate stability and balance of trade was the IMF, but they have basically self-destructed and become completely irrelevant in global economics. The only real power that the IMF had after the collapse of Bretton-Woods was the power to grant or withhold funds in a currency crisis, but they used that power so badly in the 1990’s that everyone has created these huge reserves, at which point the IMF became irrelevant.