Good point. In the United States there are two pozzolanic materials frequently used as "admixtures" in concrete: fly ash and silica fume. Fly ash is far more common. It is a beneficial byproduct of coal combustion. Being a byproduct of another industry, it is less expensive than Portland cement.

There are several reasons for adding it to mix designs. It can replace a portion of the Portland cement without sacrificing long-term strength, so it is added to reduce cost. It adds to the strength by reacting with some hydration products that do not contribute to the strength of the hardened concrete, converting those hydration products to those that do increase strength (calcium-silica hydrate for example). It reduces the permeability of the hardened concrete which improves resistance to corrosion in marine environments. It reduces the heat of hydration (which can be damaging), because it reacts less vigorously than Portland cement. There are probably other benefits, but those are the ones that come to mind.

Yes, we (Italy) often use as well flying ashes added in the concrete mix, to reduce permeability and, besides, it acts in the wet mix as a sort of lubricant and makes the pumping of the concrete much easier.

BTW Pozzolanic is way stronger (in the long run), "pure" Portland based concrete is much faster in setting, but then hits a ceiling, so where there is no particular need for fast setting (massive foundations and similar) Pozzolanic is (was) preferred (and also costed a little less).

What is called Pozzolanic cement here is anyway now essentially Portland cement 65-70% and flying ashes 30-35%, while "pure" Portland is between 90 and 95% Portland clinker and the remaining is limestone.

JFYI: https://news.ycombinator.com/item?id=29377799