I've been thinking about this question more and I've come up with a speculative possibility. Right now banks are earning their way back into profitability by playing the spread. They're paying close to zero on deposits and earning fair sums on long-term loans. Perhaps this term structure is sustainable because people are expecting little inflation in the short run but moderate inflation in the longer run, plus there is some risk on the loans. (These inflationary expectations may be changing; if you wish pretend I am writing this six months or a year ago.)
So let's say we move from zero expected short-term inflation to three percent short-term expected inflation. The nominal short rate rises to three percent and the real short rate remains more or less constant. Long rates would go up a bit but not much, since beyond the short run there is already an expectation of moderate inflation. In sum, the spread between short and long rates might narrow.
Here is the key point: from the bank's point of view, what is the correct measure of the real rate of interest? Is it defined by the nominal rate relative to the expected growth in the CPI? I doubt it. When you're near the bankruptcy or nationalization constraint, it's often nominal profits that matter (relative to fixed nominal liabilities, accounting standards, capital standards, etc.), not "real profits" defined relative to the CPI.
In sum, maybe three percent expected inflation conflicts with the desire to rapidly recapitalize banks through maintaining a wide interest rate spread. Maybe we need that zero nominal short rate or at least the Fed thinks we do.
I don't wish to push too hard on this hypothesis, it is speculative rather than confirmed by evidence. And propositions about the term structure of interest rates do not always run the way you think they will or should. I'm aware of other problems. What kind of zero profit condition is imposed on the banks? Given the odd objective function of the banks, how exactly does the Fisher effect work in the short run? Or is it imposed from without by competition from non-bank lenders? I'm not sure on these questions and they suggest possible holes in the above speculation.
I also regard this as a somewhat gruesome hypothesis. It means that "Main Street" is paying for "Wall Street" (forgive me the use of those awful terms) in at least two ways: high unemployment and inability to earn much on one's savings. Risk on the Fed balance sheet is also paying some big part of the bill, since presumably that is helping to maintain the interest rate spread.
The term structure also implies that the market is expecting rising short rates, so if the bank mess isn't cleaned up soon, heaven forbid. The spread, as a means of restoring bank profitability, won't last forever.