(Anush Kapadia, who knows this stuff much better than me, writes in with some comments on the last few posts. I accept this as a friendly amendment, and don’t disagree with any of it. I agree with particular enthusiasm with the points that we should be talking about liquidity, not money; that the the link between any quantifiable money stock and real activity had broken down by the early 1980s if not before (my point was only that it wasn’t entirely obvious until the great financial crisis); that to make sense of this stuff you need a concrete, institutionally grounded account of the financial system; and that for that, a very good place to start is Perry Mehrling’s work.)
The meaningfulness of monetary aggregates depends on the configuration of the credit system. In a world of tight banking regulations, the monetarist assumption that “there’s a stable relationship between outside money and inside money” worked fine precisely because regulations made it so. Once those regulations break down, the relationship between outside and and inside money transforms. As the mainstream understands, “the rapid pace of financial innovation in the United States has been an important reason for the instability of the relationships between monetary aggregates and other macroeconomic variables” (Bernanke, “Monetary Aggregates and Monetary Policy at the Federal Reserve: A Historical Perspective,” FRB 2006).Thus your claim that “Between 1990 and 2008, this [monetarist] story isn’t glaringly incompatible with the evidence” is not entirely true. Post-deregulation, money demand (“velocity”) became quite unmeasurable, breaking the link between the two sides of the quantity equation. “Behavior” had already changed significantly by the late 1960s, i.e. just as the monetarists were gaining the upper hand in the battle of ideas. (Note that the Fed eventually stopped measuring M3; but not everyone did: http://www.shadowstats.com/charts/monetary-base-money-supply).
Eventually, in response to this breakdown, the Fed quits its ill-conceived monetarist experiment and targets price rather quantity, specifically the Fed Funds rate. Thus “changing the stock of base money” has not been “the instrument of central banks, at least in theory, since the early 20th century.” Since the empirical and theoretical tractability of “the money supply” gave way, monetary control moved to the price of central-bank refinance, i.e. “the price of liquidity.” 
Price-based control works by acting on the leverage capacity of the balance sheets “downstream,” most immediately those in the primary dealer system. (Mehrling, New Lombard Street). Modulation of this capacity is effected through changes in the price of refinance—the bailout price—for these dealers, thereby changing their bid-ask spread. So changes in the prices of the assets in which they make markets are a key transmission mechanism to changes in interest rates.
The effect interest rates have on investment and/or consumer demand itself depends on the configuration of the credit system, i.e. how investment and consumption are financed. The price of credit might not be as important as its quantity for investment, but the former might be very important for consumption and thus aggregate demand.
So you can get a recession thanks to insufficient aggregate demand if you have a credit system that ties consumption to finance. The reason is the same as that which enables what Mehrling calls “monetary policy without sticky prices,” i.e. the leverage capacity of (in this case, consuming) balance sheets. If people are stuffed with debt, their “excess demand for money” basically represents a demand for liquidity to pay down their debts. Extra income will go first and foremost towards deleveraging rather than consumption; this of course is Richard Koo’s Minsky-flavored lesson from Japan.Given the current configuration of the system, a coordination problem of the kind referred to would mean that those with spare lending capacity can’t find those with spare borrowing capacity. Yet in sectoral terms, its only households that are truly overleveraged: government is only political so and business are relatively okay. The problem is to get the big balance sheet with the spare capacity online again; of course, that is a political problem. Boosting liquidity qua “the money supply” will simply pass through to paying down debts before it starts to affect consumption and thereby investment. In short, we might be some time, especially if we abstract away from the institutional configuration of the credit system.
 This signaled a return to pre-WWI “banking school” methods employed by the Bank of England, modulo differences in the respective credit systems: commercial paper for the trade-credit-based English system and government paper for the postwar US system. The Fed in our own period seems to be feeling its way to dealing in paper other than the government’s (QE I), something that is appropriate given the importance of non-government debt in the present system.
 Incidentally, Morris Copeland’s analogy of the credit system as an electric grid works much better than Fisher’s “currency school” vision of money as a liquid. See http://www.nber.org/books/cope52-1.