Either ways, one thing is for sure; what Americans call “Football” as in the game played with this 🏈, is definitely not Football based on how it’s played.
A game played with the ball passed around, controlled, and scored with foot, sounds more appropriate as “Football”.
The Jiu-Jitsu equivalent of arguing about whether it’s “football” or “soccer” is arguing about whether you had a Jiu-Jitsu “match” or a “fight”.