But, why use those parts?
These circuits take a lot of parts to do a job that you can do with modern high frequency stuff with a lot lower cost and parts count.
The normal point of a capacitive doubler is either to give you a voltage you need without a lot of extra parts count (often negative) or to generate a very high voltage.
My recommendation to anyone who finds themself stuck in this corner of the design space is to consider a tapped inductor converter.
I did a tapped inductor boost last year to take 3V input to 80V output (at not-much output current, I forget exactly what it was but it was mostly a bias voltage; also, the actual output voltage was DAC-set and could be quite low, so the loop dynamics were unpleasant). It was definitely annoying to wrap my head around, and very annoying to select the inductor (Würth has a nice OTS series, at the usual Würth prices; HVM would likely want a custom or semicustom design) but it just plain worked the first try and continued working through the usual stress tests and also the unusual stress tests of the Very Expensive Load™ getting itself Very Expensively Killed™ (for non-power-supply reasons). I was really happy with that converter, that kind of step-up ratio isn't easy and it just worked.
I'm a huge fan of GaN (see the current front page discussion of LiDAR), but 99% at 500 W at 12 V is hard. 3 milliohm in the inductor will eat up 100% of your budget, without accounting for any switching losses. Like most Linear parts, the LTC7820 is impressive, fits a niche, and is too expensive for most applications -- but when you need it, it's good to know it's there.
I’m with you, I’m not sure the volume or cost would be less once you factor in capacitors that are high enough quality for the application.
The datasheet mentions low profile a lot. That does make sense as one can make a flat, high quality, capacitor. Making a flat high quality inductor is harder and probably more expensive and likely consumes more volume overall. I can imagine some applications where being flat is important, like the back of a panel.
The app notes and data sheets of related parts suggest that the target application is large conversion ratios, where the duty cycle in an inductive converter is close to 0 (or 1). That forces tradeoffs, like a lower switching frequency (lower efficiency), a larger inductor (more weight and/or cost) or very short T_on for one of the FETs (lower efficiency because transition times become important). So you can use the charge pump as the first stage of a hybrid converter to get a higher system efficiency.