Also, before 1951 (and for a period of years after, although the formal break was 1951), the Fed wasn't functionally independent from govt. Because the war debt was so large, the Treasury used the Fed to press interest rates down so the debt could be paid down (it continued after 1951 because the debt was still really huge, note the similarity with your hypothetical). So the period at the end of the 50s was the first real test of Fed independence.
Again, I don't think people today understand that Fed independence is clear legally but has been more flexible in practice. Why? Because setting interest rates is inherently political. And there is an asymmetry: the incentive is always to be loose. The lesson is that there is no real way to get around political control, because the temptation is too great. I also think that policymakers should rely more heavily on macroprudential policy to take the heat out of markets (this is happening in the UK) because normal monetary policy is so asymmetric.
In theory as you approach zero smaller changes produce bigger results in an aysmptotic fashion, but in reality lenders have margin to think about.
Maybe the zero rates will take into account lenders margin, and once that is taken into effect the rates will reflect the asymptotic curve.