Working class people, such as in the teams from mill towns like Preston, Darwin and Blackburn joined the FA cup later on.
Competitive play may have started with the wealthy, but football had been played in various forms as early as the 15th century.
Wikipedia has a lot of good citations on the topic https://en.wikipedia.org/wiki/Football_(word)
As with most organically adopted words, nothing is clear cut, but there's quite a lot of evidence that the foot in football is about being on foot rather than on a horse.
Working class people, such as in the teams from mill towns like Preston, Darwin and Blackburn joined the FA cup later on.