The child was only a few months old (well under age two). I suspected it was a rogue FA, and at the time didn’t feel like challenging it. The cost/benefit of doing so didn’t seem worth it at the time.

As far as why to buy the infant a ticket (and not do infant-in-arms)?… the main reason for purchasing the kid’s seat was so that no one else was also seated in our row. Three seats across = two parents + one kid. When we had our second kid, we did infant-in-arms with them because the older kid occupied seat three in the row.