Any downside to using a charge pump for an op-amp negative supply
Yes, using a charge pump to create the negative supply for opamps can be a useful thing to do.

Often you can arrange for your internal signals to be positive, and run most things from a single supply.  Sometimes you need just a little headroom below ground.  Maybe the opamp you want to use for other reasons doesn't go all the way to the negative supply, maybe some input signal is small but ground-centered, or something else.

I have used a charge pump in such cases several times.  Nowadays just about any circuit will have a microcontroller in it, and these things often have clock outputs.  Instead of a separate oscillator for the charge pump, I usually use the clock output of the micro to drive a PNP/NPN emitter follower pair, which drives the charge pump.  You only get a couple of volts or so this way, but that's still enough to get around the headroom problem of most opamps.

I've done this where the cost of the two transistors, diodes, and caps to make the charge pump allowed using a much cheaper opamp, like the LM324.  Despite what some datasheets claim, they get flaky when their inputs get to within a diode drop of the negative supply.

As a example, take a look at the schematic of my USBProg2 PIC programmer.  You can see a charge pump in the lower right corner of page 3.  The GP0 processor pin was not otherwise used, so I assigned it to the clock out function.  This resulted in about -2.2 V, which was plenty to keep 0 V well within the proper operating range of the opamps.