While possible that local units would have played their own versions of footie during the war it would have been a big stretch to place it in the family tree in any definite way. At the time the sport if we are generous in calling it that was still by and large restricted to school grounds in the US, and far more to do with simple hazing as often as an actual game and in the old English Boy's school tradition that would spawn it's cousins Rugby and Soccer.

And while I can't say I am familiar with that piece in particular, to say the Ivies and NE Colleges grew the sport is actually to undersell it. The College's of the Mid Atlantic and New England CREATED modern Gridiron Football from the ground up with a few key outside inputs. The first game considered to be the birth of American Football then was the 1869 Rutgers Princeton game that essentially took mob games to allow upperclassmen to haze freshmen and began making a sport of it. And while the rules would have been closer to a combination of hand ball and soccer the fun of playing other schools stuck. Even then it is more that schools got the idea of playing each other that is more important than what exactly they played at this point.

Several years later Harvard then would play a series with the Canadian college McGill. One of the games was played under the new rugby rules McGill had become fond of and the Crimson men saw the appeal too. A meeting the next year with Harvard, Yale, Columbia, and several others would agree to incorporate many of these new ideas like scrimmage, down by contact, and variable point scores into their new uniform rule set. From this the Ivy League was born, and the Rules Committee under the domination of Yale Player and Coach Walter Camp would further evolve the game. Things like stoppage after being down, downs and distance, and the snap all were injected into the game in the 1880s.

It wasn't until the early 1900s though that the game would finally look familiar to a modern audience. In 1906 the predecessor to the NCAA was formed to ensure uniform play across what until then had been essentially independent conferences with the Ivy League in the East, and an emergent Big 10 out in the Mid West and other regional entities. While that same year saw the full legalization of the forward pass, until then every play was a run, lateral, or other type of toss, but nothing down the field was allowed.

So it's very much safe to say local ball games were played by regiments with house rules. But to call it in the lineage of American Football is sketchy. It's a tangled mess but once the college boys get to enjoying playing for bragging brights it's a much clearer path to the modern game.

A good look at early football I would suggest is "The Opening Kickoff" by Dave Revsine late of the Big Ten Network.

So, the answer is fairly straightforward, since American football wasn't created until after the Civil War. Although it would look quite bizarre to a viewer of the modern game, the "first" American Football game is considered to have been a match between Rutgers and Princeton, contested in 1869, and as I discussed in this very old answer, it took time for the game to spread from East Coast campuses, and the rules to standardize. So it would be quite impossible for American style-football to be getting played during the war.