Whenever I see a new product announcement saying an analog or mixed-signal component is "programmable," my first thought is simple: "in what way?" Often, it's not called out clearly or until the end of the press release.
The reason I wonder is that "programmable" is a term that can take on many different realities. Back in the day, a programmable IC (such as an op amp or filter) was one where the user could set the gain, cutoff frequency, or other characteristics via external resistors. Soon, IC vendors incorporated these resistors inside the device, so the programmability was done by jumpers on the PC board.
There were pros and cons to this setup: On the pro side, lower total cost and a simpler BOM (no need to get your own resistors, which usually had to be tightly matched for best performance). On the con side, you were limited in your settings of gain or cutoff. For most users, the choice was easy: Using internal resistors with fixed settings was a very acceptable restriction versus the benefits.
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.