Thing is, we already have a mechanism for removing money from the economy that isn't controlled by the Fed - taxation. If there's too much money, you can always increase taxes.
The difference is that taxation is subject to political forces, and politicians may not be able to pursue an unpopular policy that stabilizes the value of money. The Fed is mostly immune to short-term political needs and often pursues unpopular actions to meet its objectives.
Is there a reason that congress couldn't bestow upon the Fed some powers of taxation, while still leaving it free of the political constraints that the legislature faces?
I do not know what the law allows, but in principle a government could give such power to its central bank, and there is some precedent (the ancient Romans had a system of private tax collectors, who would basically pay the government for a kind of license to collect taxes). I am not sure there would be much of a point, since the Fed can always raise interest rates, which has the same effect on the supply of money and ultimately affects the general population in the same way (you are left with less money to save/invest/blow on parties). Taxation is somewhat more direct e.g. higher property taxes have to be paid immediately whereas a fixed rate mortgage will not suddenly become more expensive when interest rates rise, but in terms of stabilizing the value of money it does not make much of a difference one way or the other. The fact that taxes are more direct actually makes taxation the preferred tool of politicians, who typically resort to tax cuts when they are desperate for votes or propose taxes on the wealthy in order to rally support from the working class.
The difference between taxation and rate hikes is that taxes are generally progressive, while rate hikes being progressive or not depends on a lot of factors (but they can be). Taxation is a much more precise tool than rate hikes.
Again, the Fed's mission is to stabilize the value of money, and progressive tax is no better at accomplishing that objective than a flat tax. Progressive taxes may be something that people like you (and me) think are a good idea, but that is a political consideration, exactly the sort of thing the Fed is supposed to be immune to.
As for taxes being "precise," that is an interesting claim. The US tax code is not even remotely precise, in fact it is hard to know what you are supposed to pay in taxes for the first paycheck of the year (your bracket could change unexpectedly). Our income taxes are progressive, but the wealthiest Americans pay little in income tax because their "true" income comes from their investments. We have the AMT which is meant to take care of that, but the AMT has not been properly adjusted for inflation and so a bunch of middle class families are stuck paying it, even though those families also pay income tax at a higher bracket than the wealthiest households that AMT was supposed to target. Then there are the countless things a person can get credits for -- children, particular kinds of business income, charitable giving, state and local taxes, etc. -- that result in people with the same income level paying wildly different taxes from their peers. Investment income is taxed, but losses can be carried forward, and some investments are taxed in a totally different way, some can also be carried backward (see Section 1256), etc.
On some level it is "precise," but in practice it is a mess that gets more difficult to work the more "established" you are in life (bought/sold a house? got married? have a new child in your life? finally made enough to contribute to a retirement account? get ready to do a bunch of paperwork to figure out what you are supposed to pay).
Taxes, as an instrument, are multiple orders of magnitude more precise than rate hikes. Whether it is implemented well or not is an implementation detail. As a matter of essence, taxation is more precise than control of the directive rate. It is possible to wield taxes in ways that create confusing effects, or in a clear, concise way.
As for what the Fed is supposed to do, in practice, the Fed does much more than simply control the amount of money. The responsibility of the Fed is to control monetary policy and to maintain the health of the economy by doing so. Which is why they have such an important role, right now, in preventing the complete collapse of the US economy.