I don't think many engineers today would consider "programmable" to mean setting gain or frequency response via external passive components or jumpers. Programmable in today's world should exclusively mean "configurable via software." Indeed, a great many analog-mixed signal ICs that have any level of complexity beyond a couple op-amps in a package also have an I2C and internal configuration registers for adjusting functional & parametric characteristics.
As you say, there are Pro's and Con's to building an Analog chip with programmable functions. I could easily see a simple RF/Audio chip where the user could set up a number of OpAmp stages with filters, edge detectors, peak detectors and a number of other functions. Such a chip would let the "Analog Challenged" engineers with a way to quickly set up a signal conditioning chain for their sensors before turning everything over to the digital world. Same for audio or RF outputs. Sometimes you just need to tweek the final stage, but you do not need an exotic solution, just a serious of filters and buffers to improve the S/N before you blast the information into the ether.
Plus such a chip would generate interest in more people learning some of the basic Analog circuits.