Quick explanation for why a softener isn't efficient for iron. They work entirely based on ion exchange and attraction to a negatively charged surface (zeolite in the old days, synthetic resin beads now). The attraction of the cations in solution goes something like Fe3+>Fe2+>Mg2+>Ca2+>K+>Na+>H+. So if you saturated the CEC with say sodium, any ion to the left of it in that series will replace it on the resin and kick Na+ off into solution.
Iron is inefficient on a softener because iron is so much more strongly held that it takes an even larger excess of Na+ ions (than Ca and Mg do) during regenerations to get the iron off the resin. Over time if enough salt is not used, you end up getting the capacity of the softener taken up near with iron and thus decreasing your softener capacity. This is why overuse of salt/more frequent regens is critical if you use a softener for iron removal.
Besides filtering iron, a backwashing filter also provides efficient sediment filtration. Oxidant injection/sediment filtration certainly does work, but for a lot of customers it adds more things to maintain (changing filter cartridges, keeping oxidant tank filled, maintaining an injection pump, etc.)than they may want to deal with. Oxidants do have the added benefit of sanitation of your water as it is used though, so there is a benefit that just the iron filter/softener by themselves won't give you. A lot of times a carbon filter will be added to get the chlorine back out after oxidation though which usually pushes the cost of this route above just the iron filter.
I happen to like Katalox Light as an iron filter media, but there are MANY ways to get 2.5 ppm iron out of your water.