You are right about this. I did not introduce this weird biasing for the sake of that LM386; it was in the cicuit already.
I got that idea from other people's dumb designs.
However, after doing some analysis on it, prompted by experimentation and comments here, I understand why it is completely moronic---and unnecessary in, and easily removable from, those other designs where I have seen it used. For instance, in one design I was looking at, this reference network is used as an AC ground by a number of places in the circuit, such as voltage dividers in op-amp feedbacks, and attenuation voltage dividers between op-amp stages. But this is completely silly, because the real ground can be used instead, since this VREF is an AC short to the real ground anyway. Moreover, all of those couplings use a capacitor! I.e. the signal drops through the voltage divider, then through a small cap (say 0.5 uF), and then into VREF, which has two big caps (100 uF) going to the power rails. Pointless! Why not just go through the small cap, right to ground? The one or two places which actually use VREF for a voltage, rather than abusing it as a ground, can be satisified by a simple voltage divider (with no caps), which sits between the regulated power rails, and whose resistance is low enough with respect to the op-amp inputs that it's rock solid. A spare op-amp, if available, could be used as a buffer to fortify that voltage.
I understand where people are going with that: they are translating a two-supply design to a single supply circuit. The mistake is creating this middle reference network and then treating it if it were the ground in a two-supply. But it is not such a ground. Single supply requires some adjustments. The voltage divider reference is susceptible to a disturbance from low impedance couplings such as voltage sources, so you never want to couple it to such outputs. It's for "reference" only. Reference means "you can sample this voltage with a sufficiently high impedance input which won't disturb it". No such caveat applies to a real ground.
In the design I have now, I used a pair of fairly large (100K) resistors to generate a reference voltage. There are no pointless capacitors. This reference is connected to the noninverting input of an op-amp buffer (previously unused op-amp put to use). The output of this op-amp is used as a voltage reference in a couple of places (and never as a current sink or source).
The LM386 is coupled through a 300 uF resistor to the speaker, obeying the data sheet. Coupling it to the reference network would have been dumb, because even thugh the LM386 sits nicely at 4.5V, the slightest DC drift will be readily impressed upon the voltage divider (since the LM386 has a lower impedance than the divider), and therefore be fed back to earlier points in the signal chain which use the reference. This could lead to oscillations at a low frequency that isn't shunted well through the VREF caps. Or, for that matter, even oscillation at a very high frequency that also isn't shunted well through large caps due to their parasitic inductance.
David and everyone, thanks for all the comments.